Regular holonomic D-modules on skew symmetric matrices

Size: px
Start display at page:

Download "Regular holonomic D-modules on skew symmetric matrices"

Transcription

1 Regular holonomic D-modules on skew symmetric matrices Philibert ang April 6, 007 Abstract We give a classification of regular holonomic D-modules on skew-symmetric matrices attached to the linear action of the general linear group GL. Introduction The framework of the general theory of D-modules was built up by M. Sato, T. Kawai and M. Kashiwara. It is a very powerful point of view, bringing ideas from algebra and algebraic geometry to the analysis of systems of differential equations. The interest in the theory of D-modules is due to its applications in various parts of mathematics such as the representation theory, the singularity theory, the cohomology of singular spaces etc. Perhaps the first systematic use of D-modules appeared in [3]. Since then there have appeared several articles by Kashiwara and others. We should also mention the contribution of B. Malgrange. Furthermore Z. Mebkhout used the theory of D-modules to study the topology of singular varieties. Last but not least we mention the work of A. Beilinson and J. Bernstein regarding the algebraic aspect of the theory. Among the D-modules we single out a class of objects of utmost importance: the regular holonomic D-modules. One of the main problem in the D-modules theory consists in the classification of these objects. Let us point out that several authors have taken an interest in it, notably L. Boutet de Monvel [], P. Deligne, R. MacPherson and K. Vilonen [9] etc. One knows by the Riemann-Hilbert correspondence see.[5] or [0] that there is a general equivalence between the category consisting of regular holonomic D V -modules with characteristic variety Σ and the category consisting of perverse sheaves on V where V denotes a complex manifold with microsupport Σ. This gives a classification of regular holonomic D-modules theoretically, but in practice the classification of perverse sheaves is not always much simpler. A more accessible problem is as follows: given a complex manifold V on which a Lie group acts linearly with finitely many orbits V j j J ; the problem is to classify regular holonomic D V -modules whose characteristic variety is contained in the union of conormal bundles Σ := TV j V to these orbits. Closely equivalent: those that admit j J

2 a good filtration stable by infinitesimal generators of the group. These modules form a full category denoted Mod rh Σ D. In this paper we consider the action of GL C on skew-symmetric tensors. This last induces a linear action on V := Λ C, which we will think of as the space of skew-symmetric matrices. There are [ ] [ + orbits Vk 0 k ] [ ] is the integer part of : the set of rank k matrices in V. This study is done here for even which is the most interesting case. ote that here there is a natural algebra associated to this situation: the graded algebra A of polynomial coefficients differential operators acting on polynomials of the pfaffian, which is a quotient of the algebra A of SL C-invariant differential operators on V see. section 3. The main result of this paper is the theorem 8 saying that there is an equivalence of categories between the category Mod rh Σ D consisting of regular holonomic D-modules as above and the category Mod gr A consisting of graded A-modules of finite type for the Euler vector field on V. The algebra A is described simply by generators and relations see Proposition 7 thanks to a beautiful skew-capelli identity construted by R. Howe and T. Umeda see. [3, p. 59, Corollary.3.9]. This also leads to the description of the latter category as an elementary category consisting of finite dimensional vector spaces and linear maps between them satisfying certain relations Quiver category on which one can see what are the simple or indecomposable objects see. section 6. The following example is provided to illustrate the theoretical results: Example Denote pf X the pfaffian of X V, pf D its dual and the Euler vector field on V.The D V -module O V is an object in Mod rh Σ D. It is generated by an element e 0 = V such that e 0 = 0 and pf D e 0 = 0. This yields a graded A-module of finite type in Mod gr A with a basis e q where q = mk k such that pf D e 0 = 0 and satisfying the folowing system: e q = qe q q = mk, k pf X e q = e q+m, S 0 = pf D e q = m. q m + t e q m Throught the paper we assume that the reader has some familiarity with the language of D-modules. He may consult the very nice book [6] that provides a good account of an introduction to the general theory of D-modules. Finally we should recall that in precedent papers the author has obtained similar results for D-modules on C n associated to the action of the orthogonal group see. [], and on M n C the space of complex square matrices associated to the action of GL n C GL n C see. []. Definitions and preliminary results We have the following proposition. Proposition The orbits V k 0 k [ ] are simply connected.

3 Proof. Recall that the fundamental group of a homogeneous space G/H, where G is a connected group, is the component group of H. If the skew - symmetric matrices are thought as the skew bilinear forms, then a typical point is determined by a the radical of the form, which is a subspace, and b a non degenerate skew-symmetric form on the quotient. The stabilizer of this consists of all transformations which preserves the radical, and which act as an isometry of the quotient. This group is the product of a connected unipotent group with the GL of the radical and the symplectic group. Here the coefficient field is C, then this group is connected, so the orbits should be simply connected. Denote the Euler vector field on Λ C. Let M be a D-module. Definition 3 A section s in M is homogeneous if dim C C [] s <. s is homogeneous of degree λ C, if there exists j such that λ j s = 0. We recall the following useful theorem see [, Theorem.3.]: Theorem 4 Let M be a coherent D Λ C -module with a good filtration M k k Z stable by. Then i M is generated by finitely many homogeneous global sections, ii For any k, λ C, the vector space Γ Λ C [ ], M k ker λ p of homogeneous global sections of M k of degree λ is finite dimensional. Denote by G is the quotient of GL C by the kernel of its action on skew symmetric matrices. Let G := SL C be its universal covering. Definition 5 The action of the group G preserving the good filtration on a D Λ C -module M is given by an isomorphism u : p+ M p + M where p : G Λ C Λ C is the projection on Λ C, and p : G Λ C Λ C defines the action of G on Λ C. Remark 6 From [, Proposition.6.] we see that the infinitesimal action of G on M lifts to an action of its universal covering G := SL n C on M. 3 Invariant differential operators for skew-sym metric matrices In this section we describe SL C-invariant polynomial coefficients differential operators. Let us consider the action of GL C on skew-symmetric tensors. This action λ g : X gxg t g GL C naturally takes place in Λ C which we will think of as the space of skew-symmetric matrices and in particular on the algebra P Λ C of polynomials on Λ C. Thus we have p 3

4 a natural homomorphism also denoted λ from the universal enveloping algebra U gl to the algebra PD Λ C of polynomial coefficients differential operators. This homomorphism λ maps the center Z U gl to the algebra PD Λ C GL of GL -invariant differential operators with polynomial coefficients, and this restriction to the center is known to be surjective see. [3]. Thus it is natural to consider the concrete correspondence of GL -invariant differential operators and the central elements of the universal enveloping algebra U gl in more details the concrete Capelli problem see. [3, 0.4]. Here the central elements of the universal enveloping algebra U gl are the skew Capelli elements which were introduced in [3, p. 59, Remark a] to construct a skew Capelli identity for multiplicity free action of GL on skew symmetric matrices see [3, p. 59, Corollary.3.9]. ote that the explicit description of these skew Capelli elements was given by [8, p. 457, Theorem 3.] in terms of the traces of powers of a matrix defined by the standard infinitesimal generators of GL. ow we know from [3, p. 589,.3.4] see. also [4, p. 74] that the canonical generators of the algebra PD Λ C GL of GL -invariant differential operators on Λ C are the following skew Capelli operators defined with the Pfaffian: Γ Λ k := pf X I pf D I I =k where Γ Λ k has degree k as differential operator k [ ]. Here, XI and D I indicate the submatrices X I = x ij i,j I and D I = ij i,j I for I {,,, }, respectively. These GL -invariant operators are commutative, and isomorphic to the center of the universal enveloping algebra of gl C. This is discussed in [3, p , 0.4, p , and p.6 Table5. ]. 3. SL -Invariant differential operators ow denote A := PD Λ C SL the algebra of SL -invariant differential operators with polynomial coefficients on Λ C. This is sligtly larger than that of GL -invariant operators described from the Γ Λ k := pf X I pf D I. Then I =k we deduce generators for A by adding operators pf X and pf D. ote that here Γ Λ =: is the Euler vector field on Λ C. Denote J A the ideal of SL -invariant operators annihilating SL -invariant polynomials C [pf X]. Proposition 7 The algebra A of SL -invariant differential operators on Λ C is generated over C by pf X, := Γ Λ, Γ Λ,, Γ Λ, pf D subject to the following relations modulo J a [, pf X] = pf X, [, pf D] = pf D [ ] b Γ Λ k, Γ Λ l = 0 for k, l =,, c pf X pf D = + t 4

5 d pf D pf X = e [pf D, pf X] = f g h where α k := + t + Γ Λ k = α k k + t [ Γ Λ k, pf X ] { k = α k pf X [ Γ Λ k, pf D ] { k = α k Γ + Γ + kγk+. + t + + t + k + t k + t } + t } + t pf D Before going to proof of Proposition 7 we recall the explicit eigenvalues of the Capelli operators Γ Λ k. From [8, p. 463, Formula 3.0] we deduce k Γ Λ k pf X s Γ = + Γ + k Γ k + k s + t pf X s 3 where Γ z is the gamma function. As an application k =, ΓΛ = pf X pf D of this formula we note the following evaluation of the simplest Cayley type formula b-function attached to Λ C see. [8, p. 463, Corollary 3.3]: pf D pf X s = s + t pf X s. 4 Proof. of Proposition 7. Consider the following non commutative algebra B := C pf X,, Γ Λ,, ΓΛ, pf D A. 5 We show that A = B. Let V be the dual of V := Λ C and X, ξ be matrices in V V. We first show that gra is free abelian, generated by pf X, γ,, γ, pf ξ the symbols of pf X,, ΓΛ,, ΓΛ, pf D respectively. Put µ j = dz j dz j for j =,, and consider matrices X t := tµ + µ j V and ξ [t] := ξ t, t = t t µ + t j µ j V, t 0. j= j= The invariant polynomials γ k X t, ξ [t] are exactly the elementary symmetric polynomials s k t j := t i t ik, for k, j =,, This can be seen i < <i k from [8, Equations.,.4 and.6, p.45, 453.]. Let f = f X, ξ be a polynomial in X, ξ, there exists a polynomial q such that f X t, ξ [t] = q t, t,, t 6 5

6 is a polynomial in the variables t, t,, t. ote that if the variables t j are permuted one remains in the same orbit of Sp SL. So if f is an invariant polynomial, then f X t, ξ [t] is a polynomial of t and of the s k t j elementary symmetric polynomials for k, j =,, : f X t, ξ [t] = q t, s t j,, s t j. 7 Then the difference f X, ξ q pf X, γ,, γ, pf ξ 8 is a polynomial in X, ξ vanishing on the set + -affine space of X t, ξ [t]. Denote R := G X t, ξ [t] the union of orbits of points X t, ξ [t] in one of t affine spaces of V V. Assume f is invariant, then f q is invariant and vanishes on R. To see that f q vanishes everywhere i.e. gra = grb it remains to show that R is open in V V. For this purpose consider X : V V, ξ : V V skew symmetric: generically X resp. ξ is invertible then there exits g SL such that t gxg = tµ + µ j here t g denote the transpose j= of g see.[, Abstract p.9 and Theorem 4 p. 5]. Moreover the matrice Xξ is diagonalizable with distinct eigenvalues, so there exists h a symplectic transformation preserving X such that gξ t h = t t µ + t j µ j. This j= means exactly that the orbit of X t, ξ [t] contains a Zariski open dense set. So R is open in V V, and f q vanishes everywhere. Thus gra = grb. 9 ow, if P is an invariant operator of degree m 0 P A, its symbol σ P is also invariant and from 9 we get σ P = σ P pf X, γ,, γ, pf ξ. Then P can be written as the sum of an operator Q B a polynomial in the Γ Λ k s, pf X, pf D and R A an invariant operator of degree at most m : P = Q pf X,, Γ Λ,, ΓΛ, pf D + R, with deg R m. 0 By recurrence on the degree of the operator we see that P is a polynomial in the Γ Λ k s, pf X, pf D that is P B. Therefore A = B. The remaining part of the proof is devoted to desmontrating the relations a, b, c, d, e, f, g. The algebra A acts on C [pfx] the ring of 6

7 G-invariant polynomials. ow a is obvious since the pfaffian is homogeneous of degree. b holds since the GL -invariant operators commute see. [3, P. 58, 0.3 The abstract Capelli problem and p. 6, Table 5. : line 3, column 3.]. To verify c, d and e, f, g we use the relations 4, 3.. ow put A := A/J the quotient algebra of A by the ideal J. Corollary 8 A is generated by pf X,, pf D satisfying the relations [, pf X] = pf X, [, pf D] = pf D, pf X pf D = + t. Proof. Let P be in A, we decompose it into homogeneous components P = P j P j of degree j ie. [, P j] = j P j so that if j = 0 then P 0 = ϕ is j Z [ ] a polynomial in. Indeed, P 0 acts on C [pfx] then P 0 C pfx, with pfx P X = pfx. If j > 0 then pfd j P j = ψ is a polynomial in because pfd j P j is homogeneous of degree 0. Likewise if j < 0 then pfx j P j = φ is a polynomial in. Thus for any P j SL -invariant homogeneous of degree j, its class modulo J is of the form P j mod J = { pfx j φ j if j 0 pfd j ψ j if j 0 where φ j, ψ j are polynomials homogeneous of degree 0. ow put K := A pf X pf D + t A J we show that J = K. Let P be in J. Since P annihilates pfx m, m 0, then its homogeneous components P j = pfx j Q mod K if j 0 resp.pfd j Q mod K if j 0 also annihilate pfx m. This means that the polynomial in m vanishes Q m = 0 for m > j. Then we may deduced that the polynomial in λ C, Q λ = 0. Therefore Q = 0 mod K and J = K. 4 Invariant sections of D-modules on skew symmetric matrices This section is devoted to the main general argument of the paper. The idea is to show that the D Λ C -modules studied here are generated by their invariant global sections under the action of SL C. The general structure of the demonstration uses classical methods and the fact that such D Λ C -modules are essentially inverse images of modules over C by the pfaffian map. 7

8 Theorem 9 A module M in Mod rh Σ D is generated by its SL -invariant global sections. Before going to the proof of the theorem, we need the following results. 4. D-modules with support in V From now on = m. Denote V k := { X Λ C m / rank X k } the set of skew-symmetric matrices of rank k or less for 0 k m. Then V m = { X Λ C m / pf X = 0 } is the hypersurface defined by the pfaffian map pf : Λ C m C, X t. Here we study D-modules with support on V k. These modules will be used in the sequel to prove the central theorem Meromorphic sections Let be a D C -module and pf + its inverse image by the pfaffian map. ote that the transfer module D is flat over pf Λ Cm pf D C thanks to the flatness C of the pfaffian map. Thus the inverse image pf + is an exact functor. If is a regular holonomic D C -module with singularity at t = 0 t is a coordinate on C then its inverse image pf + decomposes at least as. If the operator of multiplication by t is invertible on the D C -module then the multiplication by the pfaffian pf is invertible on pf +. In particular in this case, any meromorphic section defined on Λ C m \V m extends to the whole Λ C m. Precisely, if M is a D Λ C m-module, we set M := Γ [Λ C m V m ] M = lim Hom O Λ C m I k, M where I is the defining ideal of V m the algebraic module of meromorphic sections of M with pole in the pfaffian hypersurface V m see. [6]. We have a canonical morphism M M. Proposition 0 Let be a holonomic D C -module with regular singularity at t = 0. Assume that the multiplication by t defines an automorphism of then i the multiplication by the pfaffian pf defines an automorphism of pf +, ii we have the isomorphism pf + pf +. 3 Proof. i follows from [7, Lemma., p.66] see. [7, Definition.,.3 p.64-65] and [7, Remark..,.4 p.65]. ext, recall [6] that we have an exact sequence 0 Γ [V m ] M M M where Γ [V m ] M = lim Hom O Λ C m O Λ C m/im, M is the subsheaf of M of sections annihilated by some power of I. Since pf gives a bijection on pf +, [7, Remark..,.4 and.3 p.65] asserts that H k [V m ] pf + = 0 k. Then from the previous exact sequence, we get pf + pf +. 8

9 4.. Study of pf + O t Here we describe the subquotient modules of F := O Λ C m pfx. Actually F is generated by its SL -invariant homogeneous sections e s = pf X s where s 0. From formulas 4 and 3 we get the relations k = m pf X.e s = e s+, e s = se s 4 pf D e s = Γ Λ k e s = s + t e s 5 Γ + Γ + k Γ k + k s + t e s 6 where Γ z is the gamma function Relations ote that the D Λ C -module F has + submodules denoted by F s, generated respectively by e s = pf X s s = 0,,, in O Λ C : F 0 := O Λ C F := D Λ C pf X F Consider the following quotient modules of O Λ C which will be used in the sequel: R s := O Λ C pf /F s = O Λ C pf pf := D Λ C pf X. 7 pfx by the F s /Dpf +s+. 8 The R s are generated by e +s mod e +s+, e +s mod e +s+,, e mod e +s+ homogeneous sections of degree + s, + s,, 4 respectively. Put ẽ +s k := e +s k mod e +s+ for k = 0,,s: generators ẽ +s, ẽ +s,, ẽ, pf X ẽ +s = 0 R s := ẽ +s = + s ẽ +s 9 pf X I ẽ +s = 0, I = s + for 0 s pf D I ẽ +s = 0, I = s for 0 s. thanks to the relations 4, 5, 6. Thus Char R s := T V s X. Lemma The R s := O pf /F s are modules with support on V s. 9

10 4..4 Extension In this subsection we show that any section s of the D X -module R s complementary of V s extends to the whole Λ C. in the Proposition A section s Γ Λ C m \V s, R s of the DΛ C -module R s in the complementary of V s extends to the whole Λ C m s =,, m. Proof. First, note that the hypersurface V m is smooth out of V m 4 and it is a normal variety along V m 4 smooth. Likewise the variety V s is smooth out of V s and normal along V s for s =,, m. ext, the D Λ C mmodule R s is the union of modules O Λ C ẽ +s j 0 j s such that the associated graded modules grr s is the sum of modules O T Vs Λ C ẽ +s j 0 j s. In this case the property of extension here is true for functions because V s is normal along V s s =,, m. 4. Inverse image by the pfaffian map ow let i : C Λ C m, t ω t := tdz dz + dz 3 dz dz m dz m with ωm t n! = tdz dz dz m and pf ω t = t be a section of the pfaffian map pf : Λ C m C. Denote by D = ω C = Cdz dz + dz 3 dz dz m dz m its image. We need the following lemma: Lemma 3 The line D is non characteristic for M in Mod rh Σ D i.e. T D Λ C charm T Λ C Λ C. In other words charm T Λ C D T D is a proper morphism. Proof. ote that V k D = {ω 0 } if k = m if k = 0, m D\ {ω 0 } if k = m. We show that TD Λ C TV m Λ C is contained in the zero section T Λ C Λ C. It suffices to check at the point ω 0 t = 0 which is the only point of the line D above which the characteristic variety charm has a non zero covector ξ 0 0. ote that this covector ξ 0 is parallel to dpf the conormal bundle to pfaffian variety and on the line D we have dpf X = dt 0, that is, ξ 0 / TD Λ C.. Let M be a regular holonomic D Λ C -module along charm. Since the line D is non characteristic for M then M is canonically isomorphic to the inverse image pf + i + M in the neighborhood of D. The sheaf Hom DX M, pf + i + M is constructible see. [6] and it is also locally constant on the fibers of the pfaffian map. Let u be its canonical section defined in the neighborhood of D corresponding with the isomorphism M pf + i + M which induces the identity on D. ow, thanks to the simply connectedness of the fibers of the pfaffian map see. Proposition we have the following proposition: Proposition 4 The canonical isomorphism u : M pf + i + M defined in the neighborhood of D such that i +.u = Id D, extends to Λ C m \V m 4. 0

11 Proof. The local canonical section u : M D pf + i + M D is defined out of V m 4 V V 0 the singular part of the pfaffian variety V m := pf {0}. Then we focus our attention on the orbits V m = Λ C m \pf {0} and V m = pf {0} \ V m 4 V V 0. These orbits are simply connected see. proposition : i π V m = {} ii π X m = {}. ote that the fundamental group π V m resp. π V m acts on the constructible sheaf Hom DX M, pf + i + M. This sheaf is trivial on V m = pf t and on V m = pf 0 \ V m 4 V V 0. Then the local section u : M D pf + i + M D extends globally to the union V m V m = Λ C m \ V m 4 V V 0.. Denote := i + M the restriction of the D Λ Cm-module M to the transversal line D. From Proposition 4, we deduce that the D Λ Cm-module M is isomorphic to pf + on Λ C m \V m : t 0 M Λ C m \V m pf + Λ C m \V m 0 Recall that M see section 4.. stands for the D Λ Cm-module of meromorphic sections of M defined on Λ C m \V m. According to an argument of Kashiwara, since M and pf + are regular holonomic and isomorphic out of Λ C m \V m, then their corresponding meromorphic modules are also isomorphic that is M pf +. ow let us consider the following morphism see section 4.. M M pf +. By using the basic fact that pf + pf + see relation 3 of Proposition 0 and the morphism, we deduce that there exists a morphism v : M pf + 3 which is an isomorphism out of the pfaffian hypersurface V m. Lemma 5 The image v M pf + is a D Λ Cm-module generated by its SL -invariant homogeneous global sections. ow, we are in position to prove the theorem Proof of Theorem 9 Recall F := O Λ C m pf and R s := O pf /Dpf +s+ see. section The D-module F is generated by its SL -invariant homogeneous sections e s =

12 pf s where s 0 subject to the relations 4,5 6. In particular, F has + subquotient modules denoted by R s, generated by ẽ +s, ẽ +s,, ẽ 0 s = m, and supported by V s see. Lemma. Let us denote by M M the submodule generated over D Λ C m by SL - invariant homogeneous global sections that is the module M := D Λ C m{u Γ Λ C m, M SL, dimc C [] u < }. We will see successively that the quotient module M M is supported by V s 0 s m, and the monodromy is trivial since V s \V s is simply connected. To begin with, M/ M is supported by V m : indeed M is isomorphic in Λ C m \V m to a D Λ C m-module pf + see. formula 0 and Proposition 4. We may assume that the operator of multiplication by t is invertible on such that there exists a morphism v : M pf + which is an isomorphism out of V m see. 3. The image v M is a submodule of pf + so it is generated by its SL -invariant homogeneous global sections see. Lemma 5. If σ is a SL -invariant homogeneous global section of a quotient of M then σ lifts to an invariant homogeneous global section σ of M σ Γ X, M SL, dim C C [] σ <. This means that M/ M is supported by V m. ext, if M is supported by V m, it is isomorphic out of V m 4 to a direct sum of copies of R m, then there is a morphism M R q m whose sections extend see. Proposition such that M/ M is supported by V m because the submodules of R m are also generated by their invariant homogeneous sections. In the same way by induction on k, if M is with support on V s 0 S m then there is a morphism M Rs q which is an isomorphism out of V s, such that M/ M is with support on V s because the submodules of R s are also generated by their invariant homogeneous sections. Finally, if M is supported by V 0 the Dirac module with support at the origin then the result is obvious.this ends the proof of theorem 9. 5 Equivalence of categories Recall A is the algebra of SL -invariant differential operators, J A the ideal annihilator of C [pf X] and A the quotient of A by J. Then A is generated by the 3 operators pf X,, pf D see. Corollary 8 such that [, pf X] = pf X, [, pf D] = pf D and pf X pf D = + t. We denote by Mod gr A the category of graded A-modules T of finite type such that dim C C [] u < for u T. In other words, T = T λ is a direct λ C sum of C-vector spaces T λ = ker λ p is finite dimensional equipped p with 3 endomorphisms pf X,, pf D of degree, 0,, respectively and satisfying the above relations with λ being a nilpotent operator on each T λ. Let us recall that Mod rh Σ D stands for the category of regular holonomic D Λ C m-

13 modules whose characteristic variety is contained in Σ. If M is an object in the category Mod rh Σ D, denote by Ψ M the submodule of Γ Λ C m, M consisting of SL -invariant homogeneous global sections u of M such that dim C C [] u <. Recall that Theorem 4 Ψ M λ := [Ψ M] [ ] ker λ p is the C-vector space of homogeneous global sections of p degree λ of Ψ M and Ψ M = λ C Ψ M λ. Then Ψ M is an object in the category Mod gr A. Indeed, let σ,, σ p be a finite family of homogeneous invariant global sections generating the D Λ Cm-module M see Theorem 9, we can see that the family σ,, σ p generates also Ψ M as an A-module: In fact, if σ = p j= q j X, D σ j is an invariant section of M q j Γ Λ C m, D Λ C m, denote by qj the average of q j over SU n C compact maximal subgroup of SL, then q j A. Let f j = f j pf X,, pf D be the class of q j modulo J that is f j A, then we also have σ = p j= q j σ j = p j= f j σ j. Conversely, if T is an object in the category Mod gr A, one associates to it the D Λ C m-module Φ T = M 0 T 4 where M 0 := D/J is a D, A-module. Then Φ T is an object in the category Mod rh Σ D. Thus, we have defined two functors A Ψ : Mod rh Σ D Mod gr A, Φ : Mod gr A Mod rh Σ D. 5 We need the following lemmas: Lemma 6 The canonical morphism T ΨΦ T, t t 6 is an isomorphism, and defines an isomorphism of functors Id Mod gr A Ψ Φ. Proof. As above M 0 := D/J. Denote by ε the class of D modulo J the canonical generator of M 0. Let h D, denote by h A its average on SU C and by ϕ the class modulo J that is ϕ A. Since ε is SL -invariant, we get hε = hε = εϕ. Moreover, we have hϕ = 0 if and only if h J, in other words ϕ = 0. Therefore the average operator over SU C D A, h h induces a surjective morphism of A-modules v : M 0 A. More generally, for any A-module T in the category Mod gr A the morphism v T is a surjective map v T : M 0 T A T = T 7 A A 3

14 which is the left inverse of the morphism u T : T M 0 T, t ε t 8 that is v T ε T = v ε = T. This means that the morphism u T is injective. ext, the image of u T is exactly the set of invariant sections of M 0 A T = Φ T that is ΨΦT : indeed if σ = p h i t i is an invariant section in M 0 A T, we may replace each h i by its average h i A, then we get that is p i= σ = A p h i t i = ε i= i= p h i t i ε T 9 i= h i t i T. Therefore the morphism u T is an isomorphism from T to Ψ Φ T and defines an isomorphism of functors. Lemma 7 The canonical morphism w : Φ Ψ M M 30 is an isomorphism and defines an isomorphism of functors Φ Ψ Id Mod rh Σ D. Proof. As in the Theorem 9 the D Λ Cm-module M is generated by a finite family of invariant sections σ i i=,,p Ψ M so that the morphism w is surjective. Anyway w is injective. Indeed let Q be the kernel of the morphism w : Φ Ψ M M. The D Λ Cm-module Q is also generated by its invariant sections that is by Ψ Q. Then we get Ψ Q Ψ [Φ Ψ M] = Ψ M 3 where we used Ψ Φ = Id Mod gr A see the previous Lemma 6. Since the morphism Ψ M M is injective Ψ M Γ Λ C m, M, we obtainψ Q = 0. Therefore Q = 0 because Ψ Q generates Q.. This section ends by the following theorem established by means of the previous lemmas: Theorem 8 The functors Φ and Ψ induce equivalence of categories Mod rh Σ D Mod gr A. 3 6 Classification of finite type graded A-modules This section consists in the classification of objects in the category Mod gr A. A graded A-module T in Mod gr A defines an infinite diagram consisting of 4

15 finite dimensional vector spaces T λ with λ being a nilpotent operator on each T λ, λ C and linear maps between them deduced from pf X,, pf D: pfx T λ T λ+ pfd satisfying the following λ T λ T λ, pf X pf D = pf D pf X = 6. Examples + t t, We describe graded A-modules of finite type and the corresponding diagrams associated to D V -modules O V, B {0} V, O V pfx /O V, which are regular holonomic with characteristic variety contained in Σ. Example 9 The D V -module O V is generated by an element e 0 = V such that e 0 = 0 and pf D e 0 = 0. Then its associated graded A-module has a basis e q where q = mk k such that pf D e 0 = 0 and satisfying the system: e q = qe q q = mk, k pf X e q = e q+m, S 0 = pf D e q = m. 34 q m + t e q m Since pf D e 0 = 0 i.e pf D T 0 = 0, the arrows at the left of T 0 in the diagram vanish. Example 0 The D V -module B {0} V is generated by an element e m such that e m = me m and pf X e m = 0. Its associated A-module has a basis e q where q = m mk k such that pf X e m = 0 satisfying the system: e q = qe q q = m mk, k pf D e q = e q m, S = pf X e q = m 35 q m + t + e q+m Since pf X e m = 0 i.e pf X T m = 0, the arrows at the right of T m in the diagram vanish. Example The D V -module O V /O V is generated by an element e m = pfx pfx mod O V such that e m = me m and pf X e m = e 0. Then its associated A-module has a basis e q where q = mk k satisfying a system as S. 5

16 Actually diagrams studied are completely determined by a finite subset of objects and arrows. Indeed a For σ C/ Z, denote by T σ T the submodule T σ = T λ. Then λ=σmod Z T is generated by the finite direct sum of T σ s T = T σ = T λ. 36 σ ɛ C/ Z σ ɛ C/ Z λ=σmod Z b If σ 0 mod Z λ = σmod Z, then the linear maps pf X and pf D are bijective. Therefore T σ is completely determined by one element T λ equipped with the nilpotent action of λ. c If σ = 0 mod mz λ = σmodmz with = m, then T σ is completely determined by one diagrams of m elements T m m pfx pfd pfx T m m T m T 0 37 pfd In the other degrees pf X or pf D are bijective. Indeed, we have T 0 pf X k T 0 T mk and T m m pf D k T m m T m +km k thanks to the relations pf X pf D = m m + t and pf D pf X = m m + t +. The operator pf X pf D resp. pf D pf X on T λ has only one eigenvalue λ m λ m + λ n + 4 λ m + m resp. λ m + λ m + 3 λ m + m so that the equation pf X pf D = m m + t resp.pf D pf X = m m + t + has a unique solution of eigenvalue λ if λ is not a critical value. Here λ = 0, m, 4m,, m m or λ = m,, m m thus it is always the case.. Aknowledgements: The author expresses his sincere thanks to Professor Louis Boutet de Monvel for many helpful discussions, to Professor Roger Howe for kindly pointing out his paper with T. Umeda [3]. This work was done during the author s stay at Institut de Mathématiques de Jussieu and at Max- Planck-Institut für Mathematik, he is gratefull to these institutions for their hospitality and support. References [] L. Boutet de Monvel, D-modules holonômes réguliers en une variable, Mathématiques et Physique, Séminaire de L ES, Progr.Math., 37 6

17 Birkhäuser Boston, MA, 97-98, [] R. Gow, T.J Laffey, Pairs of alternating forms and products of two skewsymmetric matrices, Linear algebra Appl., 63, [3] R. Howe, T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity free actions, Math. Ann., 90, [4] M. Itoh, A Cayley-Hamilton theorem for skew Capelli elements. J. Algebra, vol no., [5] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci , [6] M. Kashiwara, D-modules and Microlocal calculus, Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs, AMS, vol [7] M. Kashiwara, T. Kawai, On the characteristic variety of a holonomic system with regular singularities, Adv. in Math , no., [8] K. Kinoshita, M. Wakayama, The explicit Capelli identities for skew symmetric matrices, Proc. Edinburgh. math. Soc., [9] R. Macpherson, K. Vilonen, Elementary construction of perverse sheaves. Invent. Math [0] Z. Mebkhout, Une autre équivalence de catégories, Compositio Math , [] P. ang, D-modules associated to the group of similitudes, Publ. Res. Inst. Math. Sci , [] P. ang, D-modules associated to determinantal singularities, Proc. Japan Acad. Ser. A Math. Sci , [3] M. Sato, T. Kawai, M. Kashiwara, Microfunctions and pseudo differential equations. Hyperfunctions and pseudo diffrential equations Proc. Conf., Katata 97; dedicated to memory of André Martineau 97 pp Lecture otes in Math., Vol. 87, Springer Berlin 973. Philibert ang Max-Planck Institute for Mathematics Vivatsgasse 7, D-53 Bonn, Germany adress: pnang@mpim-bonn.mpg.de 7

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of D-modules Mainz July 9, 2018 1 The

More information

Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)]. Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

More information

1. Algebraic vector bundles. Affine Varieties

1. Algebraic vector bundles. Affine Varieties 0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

More information

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

More information

LECTURE 4.5: SOERGEL S THEOREM AND SOERGEL BIMODULES

LECTURE 4.5: SOERGEL S THEOREM AND SOERGEL BIMODULES LECTURE 4.5: SOERGEL S THEOREM AND SOERGEL BIMODULES DMYTRO MATVIEIEVSKYI Abstract. These are notes for a talk given at the MIT-Northeastern Graduate Student Seminar on category O and Soergel bimodules,

More information

The Grothendieck-Katz Conjecture for certain locally symmetric varieties

The Grothendieck-Katz Conjecture for certain locally symmetric varieties The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-

More information

Algebraic Geometry Spring 2009

Algebraic Geometry Spring 2009 MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

More information

MAT 5330 Algebraic Geometry: Quiver Varieties

MAT 5330 Algebraic Geometry: Quiver Varieties MAT 5330 Algebraic Geometry: Quiver Varieties Joel Lemay 1 Abstract Lie algebras have become of central importance in modern mathematics and some of the most important types of Lie algebras are Kac-Moody

More information

Construction of M B, M Dol, M DR

Construction of M B, M Dol, M DR Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant

More information

LECTURE 4: SOERGEL S THEOREM AND SOERGEL BIMODULES

LECTURE 4: SOERGEL S THEOREM AND SOERGEL BIMODULES LECTURE 4: SOERGEL S THEOREM AND SOERGEL BIMODULES DMYTRO MATVIEIEVSKYI Abstract. These are notes for a talk given at the MIT-Northeastern Graduate Student Seminar on category O and Soergel bimodules,

More information

Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

More information

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0

More information

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle on a general hypersurface

More information

On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem

On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013

More information

Tamagawa Numbers in the Function Field Case (Lecture 2)

Tamagawa Numbers in the Function Field Case (Lecture 2) Tamagawa Numbers in the Function Field Case (Lecture 2) February 5, 204 In the previous lecture, we defined the Tamagawa measure associated to a connected semisimple algebraic group G over the field Q

More information

Notes on p-divisible Groups

Notes on p-divisible Groups Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

More information

arxiv:math/ v1 [math.ag] 24 Nov 1998

arxiv:math/ v1 [math.ag] 24 Nov 1998 Hilbert schemes of a surface and Euler characteristics arxiv:math/9811150v1 [math.ag] 24 Nov 1998 Mark Andrea A. de Cataldo September 22, 1998 Abstract We use basic algebraic topology and Ellingsrud-Stromme

More information

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1

More information

Math 797W Homework 4

Math 797W Homework 4 Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

More information

SYSTEMS OF MEROMORPHIC MICRODIFFERENTIAL EQUATIONS

SYSTEMS OF MEROMORPHIC MICRODIFFERENTIAL EQUATIONS SINGULARITIES AND DIFFERENTIAL EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 33 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1996 SYSTEMS OF MEROMORPHIC MICRODIFFERENTIAL EQUATIONS ORLAND

More information

Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

More information

A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1

A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection

More information

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 18. Category O for rational Cherednik algebra We begin to study the representation theory of Rational Chrednik algebras. Perhaps, the class of representations

More information

Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

More information

REPRESENTATION THEORY, LECTURE 0. BASICS

REPRESENTATION THEORY, LECTURE 0. BASICS REPRESENTATION THEORY, LECTURE 0. BASICS IVAN LOSEV Introduction The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite

More information

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

More information

On the modular curve X 0 (23)

On the modular curve X 0 (23) On the modular curve X 0 (23) René Schoof Abstract. The Jacobian J 0(23) of the modular curve X 0(23) is a semi-stable abelian variety over Q with good reduction outside 23. It is simple. We prove that

More information

8 Perverse Sheaves. 8.1 Theory of perverse sheaves

8 Perverse Sheaves. 8.1 Theory of perverse sheaves 8 Perverse Sheaves In this chapter we will give a self-contained account of the theory of perverse sheaves and intersection cohomology groups assuming the basic notions concerning constructible sheaves

More information

The V -filtration and vanishing and nearby cycles

The V -filtration and vanishing and nearby cycles The V -filtration and vanishing and nearby cycles Gus Lonergan Disclaimer: We will work with right D-modules. Any D-module we consider will be at least coherent. We will work locally, choosing etale coordinates

More information

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

More information

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive

More information

Coherent sheaves on elliptic curves.

Coherent sheaves on elliptic curves. Coherent sheaves on elliptic curves. Aleksei Pakharev April 5, 2017 Abstract We describe the abelian category of coherent sheaves on an elliptic curve, and construct an action of a central extension of

More information

LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)

LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups

More information

REPRESENTATION THEORY. WEEKS 10 11

REPRESENTATION THEORY. WEEKS 10 11 REPRESENTATION THEORY. WEEKS 10 11 1. Representations of quivers I follow here Crawley-Boevey lectures trying to give more details concerning extensions and exact sequences. A quiver is an oriented graph.

More information

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ

More information

12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.

12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End

More information

Geometry 9: Serre-Swan theorem

Geometry 9: Serre-Swan theorem Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

More information

Cohomology and Base Change

Cohomology and Base Change Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)

More information

Theta divisors and the Frobenius morphism

Theta divisors and the Frobenius morphism Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following

More information

Noetherian property of infinite EI categories

Noetherian property of infinite EI categories Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result

More information

A GEOMETRIC JACQUET FUNCTOR

A GEOMETRIC JACQUET FUNCTOR A GEOMETRIC JACQUET FUNCTOR M. EMERTON, D. NADLER, AND K. VILONEN 1. Introduction In the paper [BB1], Beilinson and Bernstein used the method of localisation to give a new proof and generalisation of Casselman

More information

On the geometric Langlands duality

On the geometric Langlands duality On the geometric Langlands duality Peter Fiebig Emmy Noether Zentrum Universität Erlangen Nürnberg Schwerpunkttagung Bad Honnef April 2010 Outline This lecture will give an overview on the following topics:

More information

The Maslov Index. Seminar Series. Edinburgh, 2009/10

The Maslov Index. Seminar Series. Edinburgh, 2009/10 The Maslov Index Seminar Series Edinburgh, 2009/10 Contents 1 T. Thomas: The Maslov Index 3 1.1 Introduction: The landscape of the Maslov index............... 3 1.1.1 Symplectic vs. projective geometry...................

More information

Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

More information

Characters of the Negative Level Highest-Weight Modules for Affine Lie Algebras

Characters of the Negative Level Highest-Weight Modules for Affine Lie Algebras IMRN International Mathematics Research Notices 1994, No. 3 Characters of the Negative Level Highest-Weight Modules for Affine Lie Algebras Masaki Kashiwara and Toshiyuki Tanisaki 0 Introduction Our main

More information

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n)

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) VERA SERGANOVA Abstract. We decompose the category of finite-dimensional gl (m n)- modules into the direct sum of blocks, show that

More information

1 Hermitian symmetric spaces: examples and basic properties

1 Hermitian symmetric spaces: examples and basic properties Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................

More information

Chern classes à la Grothendieck

Chern classes à la Grothendieck Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

More information

Duality, Residues, Fundamental class

Duality, Residues, Fundamental class Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class

More information

Micro-support of sheaves

Micro-support of sheaves Micro-support of sheaves Vincent Humilière 17/01/14 The microlocal theory of sheaves and in particular the denition of the micro-support is due to Kashiwara and Schapira (the main reference is their book

More information

BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt, CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

More information

Generic section of a hyperplane arrangement and twisted Hurewicz maps

Generic section of a hyperplane arrangement and twisted Hurewicz maps arxiv:math/0605643v2 [math.gt] 26 Oct 2007 Generic section of a hyperplane arrangement and twisted Hurewicz maps Masahiko Yoshinaga Department of Mathematice, Graduate School of Science, Kobe University,

More information

Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

More information

arxiv: v1 [math.gr] 8 Nov 2008

arxiv: v1 [math.gr] 8 Nov 2008 SUBSPACES OF 7 7 SKEW-SYMMETRIC MATRICES RELATED TO THE GROUP G 2 arxiv:0811.1298v1 [math.gr] 8 Nov 2008 ROD GOW Abstract. Let K be a field of characteristic different from 2 and let C be an octonion algebra

More information

A Gauss-Bonnet theorem for constructible sheaves on reductive groups

A Gauss-Bonnet theorem for constructible sheaves on reductive groups A Gauss-Bonnet theorem for constructible sheaves on reductive groups V. Kiritchenko 1 Introduction In this paper, we prove an analog of the Gauss-Bonnet formula for constructible sheaves on reductive groups.

More information

2. D-MODULES AND RIEMANN-HILBERT

2. D-MODULES AND RIEMANN-HILBERT 2. D-MODULES AND RIEMANN-HILBERT DONU ARAPURA The classical Riemann-Hilbert, or Hilbert s 21st, problem asks whether every representation of the punctured complex plane comes from a system of differential

More information

1 Notations and Statement of the Main Results

1 Notations and Statement of the Main Results An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main

More information

1 Moduli spaces of polarized Hodge structures.

1 Moduli spaces of polarized Hodge structures. 1 Moduli spaces of polarized Hodge structures. First of all, we briefly summarize the classical theory of the moduli spaces of polarized Hodge structures. 1.1 The moduli space M h = Γ\D h. Let n be an

More information

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Benson Farb and Mark Kisin May 8, 2009 Abstract Using Margulis s results on lattices in semisimple Lie groups, we prove the Grothendieck-

More information

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON THREE DIMENSIONAL HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON THREE DIMENSIONAL HYPERSURFACES ARITHMETICALLY COHEN-MACAULAY BUNDLES ON THREE DIMENSIONAL HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle

More information

On the closures of orbits of fourth order matrix pencils

On the closures of orbits of fourth order matrix pencils On the closures of orbits of fourth order matrix pencils Dmitri D. Pervouchine Abstract In this work we state a simple criterion for nilpotentness of a square n n matrix pencil with respect to the action

More information

Math 210C. The representation ring

Math 210C. The representation ring Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let

More information

Thus we get. ρj. Nρj i = δ D(i),j.

Thus we get. ρj. Nρj i = δ D(i),j. 1.51. The distinguished invertible object. Let C be a finite tensor category with classes of simple objects labeled by a set I. Since duals to projective objects are projective, we can define a map D :

More information

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

More information

A Leibniz Algebra Structure on the Second Tensor Power

A Leibniz Algebra Structure on the Second Tensor Power Journal of Lie Theory Volume 12 (2002) 583 596 c 2002 Heldermann Verlag A Leibniz Algebra Structure on the Second Tensor Power R. Kurdiani and T. Pirashvili Communicated by K.-H. Neeb Abstract. For any

More information

Another proof of the global F -regularity of Schubert varieties

Another proof of the global F -regularity of Schubert varieties Another proof of the global F -regularity of Schubert varieties Mitsuyasu Hashimoto Abstract Recently, Lauritzen, Raben-Pedersen and Thomsen proved that Schubert varieties are globally F -regular. We give

More information

A Note on Dormant Opers of Rank p 1 in Characteristic p

A Note on Dormant Opers of Rank p 1 in Characteristic p A Note on Dormant Opers of Rank p 1 in Characteristic p Yuichiro Hoshi May 2017 Abstract. In the present paper, we prove that the set of equivalence classes of dormant opers of rank p 1 over a projective

More information

LECTURE 11: SOERGEL BIMODULES

LECTURE 11: SOERGEL BIMODULES LECTURE 11: SOERGEL BIMODULES IVAN LOSEV Introduction In this lecture we continue to study the category O 0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing

More information

Lecture 6: Etale Fundamental Group

Lecture 6: Etale Fundamental Group Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and

More information

LECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

LECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment. LECTURE 3 MATH 261A LECTURES BY: PROFESSOR DAVID NADLER PROFESSOR NOTES BY: JACKSON VAN DYKE Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

More information

Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

More information

THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

More information

DERIVED CATEGORIES: LECTURE 4. References

DERIVED CATEGORIES: LECTURE 4. References DERIVED CATEGORIES: LECTURE 4 EVGENY SHINDER References [Muk] Shigeru Mukai, Fourier functor and its application to the moduli of bundles on an abelian variety, Algebraic geometry, Sendai, 1985, 515 550,

More information

Lecture 4 Super Lie groups

Lecture 4 Super Lie groups Lecture 4 Super Lie groups In this lecture we want to take a closer look to supermanifolds with a group structure: Lie supergroups or super Lie groups. As in the ordinary setting, a super Lie group is

More information

CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS. Contents. 1. The ring K(R) and the group Pic(R)

CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS. Contents. 1. The ring K(R) and the group Pic(R) CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS J. P. MAY Contents 1. The ring K(R) and the group Pic(R) 1 2. Symmetric monoidal categories, K(C), and Pic(C) 2 3. The unit endomorphism ring R(C ) 5 4.

More information

PERVERSE SHEAVES ON A TRIANGULATED SPACE

PERVERSE SHEAVES ON A TRIANGULATED SPACE PERVERSE SHEAVES ON A TRIANGULATED SPACE A. POLISHCHUK The goal of this note is to prove that the category of perverse sheaves constructible with respect to a triangulation is Koszul (i.e. equivalent to

More information

Math 249B. Geometric Bruhat decomposition

Math 249B. Geometric Bruhat decomposition Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique

More information

GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui

GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY Sampei Usui Abstract. This article is a geometric application of polarized logarithmic Hodge theory of Kazuya Kato and Sampei Usui. We prove generic Torelli

More information

Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

More information

Math 121 Homework 4: Notes on Selected Problems

Math 121 Homework 4: Notes on Selected Problems Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W

More information

Derived Canonical Algebras as One-Point Extensions

Derived Canonical Algebras as One-Point Extensions Contemporary Mathematics Derived Canonical Algebras as One-Point Extensions Michael Barot and Helmut Lenzing Abstract. Canonical algebras have been intensively studied, see for example [12], [3] and [11]

More information

Introduction to Chiral Algebras

Introduction to Chiral Algebras Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument

More information

ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS

ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,

More information

ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY

ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up

More information

Symplectic varieties and Poisson deformations

Symplectic varieties and Poisson deformations Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the

More information

Exercises on chapter 1

Exercises on chapter 1 Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G

More information

PBW for an inclusion of Lie algebras

PBW for an inclusion of Lie algebras PBW for an inclusion of Lie algebras Damien Calaque, Andrei Căldăraru, Junwu Tu Abstract Let h g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module

More information

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

More information

Groups of Prime Power Order with Derived Subgroup of Prime Order

Groups of Prime Power Order with Derived Subgroup of Prime Order Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*

More information

Determinant lines and determinant line bundles

Determinant lines and determinant line bundles CHAPTER Determinant lines and determinant line bundles This appendix is an exposition of G. Segal s work sketched in [?] on determinant line bundles over the moduli spaces of Riemann surfaces with parametrized

More information

TitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2)

TitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2) TitleOn manifolds with trivial logarithm Author(s) Winkelmann, Jorg Citation Osaka Journal of Mathematics. 41(2) Issue 2004-06 Date Text Version publisher URL http://hdl.handle.net/11094/7844 DOI Rights

More information

ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS

ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical

More information

A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander

A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander During the first three days of September, 1997, I had the privilege of giving a series of five lectures at the beginning of the School on Algebraic

More information

VANISHING OF CERTAIN EQUIVARIANT DISTRIBUTIONS ON SPHERICAL SPACES

VANISHING OF CERTAIN EQUIVARIANT DISTRIBUTIONS ON SPHERICAL SPACES VANISHING OF CERTAIN EQUIVARIANT DISTRIBUTIONS ON SPHERICAL SPACES AVRAHAM AIZENBUD AND DMITRY GOUREVITCH Abstract. We prove vanishing of z-eigen distributions on a spherical variety of a split real reductive

More information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12 MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

More information

Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005)

Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005) Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005) U. Bunke April 27, 2005 Contents 1 Abelian varieties 2 1.1 Basic definitions................................. 2 1.2 Examples

More information

Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and

More information

Isogeny invariance of the BSD conjecture

Isogeny invariance of the BSD conjecture Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p

More information

Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

More information

IVAN LOSEV. Lemma 1.1. (λ) O. Proof. (λ) is generated by a single vector, v λ. We have the weight decomposition (λ) =

IVAN LOSEV. Lemma 1.1. (λ) O. Proof. (λ) is generated by a single vector, v λ. We have the weight decomposition (λ) = LECTURE 7: CATEGORY O AND REPRESENTATIONS OF ALGEBRAIC GROUPS IVAN LOSEV Introduction We continue our study of the representation theory of a finite dimensional semisimple Lie algebra g by introducing

More information