# LECTURE 3: TENSORING WITH FINITE DIMENSIONAL MODULES IN CATEGORY O

Size: px
Start display at page:

Transcription

1 LECTURE 3: TENSORING WITH FINITE DIMENSIONAL MODULES IN CATEGORY O CHRISTOPHER RYBA Abstract. These are notes for a seminar talk given at the MIT-Northeastern Category O and Soergel Bimodule seminar (Autumn 2017). Contents 1. Goals 1 2. Review of Notation 1 3. Tensor Product with Finite-Dimensional Modules 2 4. Projective Functors, Translation Functors, and Equivalences of Categories 2 5. Projections to the Walls and Reflection Functors 5 6. Projective Objects in Category O 6 1. Goals The purpose of this document is to discuss the operation of tensoring with a finite dimensional module in the category O associated to a finite-dimensional semisimple Lie algebra g. In particular we will investigate the behaviour of certain functors between infinitesimal blocks of the category that arise from these tensor products, and relations to projective objects in an infinitesimal block O χ. We will use the following notation: 2. Review of Notation A finite-dimensional semisimple Lie algebra g over C. A triangular decomposition g = n h n +. In particular h is (a choice of) a Cartan subalgebra, and b = h n + is the (positive) Borel subalgebra. Simple roots α i, with roots R and positive roots R + and negative roots R. We write ρ = 1 2 α R + α. We write Λ for the weight lattice. We write W for the Weyl group associated to g, which acts on h in the usual way, and also via the dotted action: w v = w(v + ρ) ρ (w W ). We write Stab W (λ) for the stabiliser of λ with respect to the usual action. W is generated by the simple reflections s i which correspond to the simple roots α i. The category of finitely generated g-modules that are h-semisimple and locally nilpotent for the action of n + is denoted O. It has a duality operation M M. Verma modules of highest weight λ, (λ). These have unique simple quotient L(λ). Both of these belong to O. We also have dual Verma modules (λ) = (λ). There is a decomposition of categories O = λ h /W O λ. The quotient is by the dotted action of W. The summand O λ has simple objects indexed by L(w λ) for w W. There are W/Stab W (λ + ρ) of them. Date: September 24,

2 3. Tensor Product with Finite-Dimensional Modules Henceforth, V is a finite dimensional g-module. Proposition 3.1. Tensoring with V defines a functor O O. This functor is exact, and has left and right adjoints both equal to tensoring with V. Proof. It was shown in the previous lecture that tensoring with a finite dimensional module takes modules in category O to modules in category O. To prove adjointness properties, we use the property that Hom C (V, M) = V M (which requires V to be finite dimensional). We argue that: Hom C (M C V, N) = Hom C (M, Hom C (V, N)) = Hom C (M, N V ) Hom C (N, M C V ) = Hom C (N, Hom C (V, M)) = Hom C (M V, N) Certainly, the tensor-hom adjunction guarantees this equality for the underlying vector spaces. It remains to check that this map respects g-module structure. This is well known and easy to check. Proposition 3.2. Let ν i (i {1, 2,, dim(v )}) be the weights of V with multiplicity, ordered such that if ν i ν j in the usual partial order on h, then i j. Then M = (λ) V has a filtration M = M 0 M 1 M dim(v ) = 0 such that M i /M i+1 = (λ + νi ). Proof. We use the tensor-hom adjunction, and the fact that (λ) = U(g) U(b) C λ (where C λ is the one dimensional representation of b associated to the weight λ). Let N be a g-module. Hom g (V (λ), N) = Hom g (V (U(g) U(b) C λ ), N) = Hom g (U(g) U(b) C λ, V N) = Hom b (C λ, V N) = Hom b (V C λ, N) = Hom g (U(g) U(b) (V C λ ), N) Now we observe that V C λ is filtered as a b-module by W j = Span({v 1, v 2,, v j }) (j = 1, 2,, dim(v )). Note that W j /W j+1 is one dimensional of weight ν j. Since U(g) is free over U(b) (by the PBW theorem), tensoring up to U(g) is an exact functor, so the filtration of V C λ passes to a filtration of V (λ) with subquotients U(g) U(b) C λ+νi = (λ + ν i ). This proves the proposition. Remark 3.3. Recall that O = χ h /W O χ (decomposition into infinitesimal blocks). It was shown in the previous talk that O χ has enough projectives, and that projectives admit a filtration by Verma modules. This makes O χ into a highest-weight category, where the standard objects are Verma modules. 4. Projective Functors, Translation Functors, and Equivalences of Categories Definition 4.1. Let µ h /W and pr µ : O O µ be the functor projecting onto the summand O µ. A projective functor is a functor of the form pr µ (V ι λ ( )) : O λ O µ, where ι λ is the inclusion O λ O. Remark 4.2. Observe that pr µ (V ι λ ( )) is biadjoint to pr λ (V ι µ ( )). This is immediate from Proposition 3.1. These functors will allows us to prove equivalences of infinitesimal blocks associated to dominant integral weights, and describe the structure of the categories in the non-dominant integral case. Proposition 4.3. If M M is the duality operation defined in the previous lecture, then we have that pr µ (V ι(m )) = pr µ (V ι λ (M)). Proof. Firstly, note that duality preserves infinitesimal blocks. To see this, it is enough to know that simple objects are self dual under duality (shown last lecture), then the infinitesimal blocks are generated by successive extensions of the simple objects in the infinitesimal block. Then, the claim reduces to showing that V M = (V M). It is easy to see that (V M) = V M, so it is enough to show that finite dimensional representations are self-dual. The category of finite-dimensional representations of g is semisimple, so it is enough to see that V and V have the same character, which is trivial. 2

3 The following statement is immediate from the Proposition 3.2 and the characterisation of the infinitesimal blocks O µ. Corollary 4.4. The object pr µ (V (w λ)) admits a filtration by (w λ + ν i ), where w λ + ν i W µ (each with multiplicity 1). Remark 4.5. In view of Proposition 4.3, Corollary 4.4 holds with Verma modules replaced by dual Verma modules. Proposition 4.6. Let λ, µ P be such that λ, µ + ρ, λ µ are all dominant, and w W. Let V be the finite dimensional simple g-module of highest weight λ µ. We have that pr µ (V (w λ)) = (w µ), and also that pr λ (V (w µ)) is filtered by modules of the form (w λ), where w wstab W (µ + ρ) (each with multiplicity 1). Proof. We consider pr µ (V (w λ)), using the Proposition 3.2 to find all candidates for terms arising in a filtration by Verma modules. The weights of V are precisely the negatives of those in V. For each i, obtain a Verma module of highest weight w λ ν i provided that w λ ν i W µ (and none otherwise). We must find w and ν i such that w λ ν i = w µ (where w W ). We write the dotted action in terms of the usual one, and act by w 1 to get λ+ρ w 1 ν i = w 1 w (µ+ρ). Since λ µ is dominant, λ µ w 1 ν i (recall that ν i is a weight of V, which has highest weight λ µ), and equality holds if and only if w 1 ν i is a the highest weight λ µ. Similarly, µ + ρ w 1 w (µ + ρ), and equality holds if and only if w 1 w Stab W (µ + ρ). We use these inequalities to get: λ + ρ w 1 ν i λ + ρ (λ µ) = µ + ρ w 1 w (µ + ρ) The condition λ + ρ w 1 ν i = w 1 w (µ + ρ) holds if and only if both inequalities were equalities. This gives ν i = w(λ µ) and w wstab W (µ + ρ). Note that the highest weight space of V is one-dimensional, and therefore the weight space of weight w(λ µ) is also one-dimensional as it is in the same Weyl group orbit as the highest weight space; this means that there is exactly one choice of i satisfying the condition. We obtain a one-term filtration by (w λ w(λ µ)) = (w µ). Hence, pr µ (V (w λ)) = (w µ). To see the filtration statement, we perform an analogous analysis, but with ν i replaced by ν i. We need to consider w µ+ν i = w λ, which becomes w 1 w(µ+ρ)+w 1 ν i = λ+ρ. The inequalities w 1 w(µ+ρ) µ+ρ and w 1 ν i λ µ give: w 1 w(µ + ρ) + w 1 νi µ + ρ + w 1 νi µ + ρ + (λ µ) = λ + ρ We have equality only when w 1 w Stab W (µ + ρ) and ν i = w (λ µ). The first condition is equivalent to w 1 w Stab W (µ+ρ), so w wstab W (µ+ρ). The second condition determines the terms in the filtration: (w µ + ν i ) = (w µ + w (λ µ)) = (w µ + w (λ µ)) = (w λ) Conversely, each w wstab W (µ + ρ) gives rise to a term corresponding to ν i = w (λ ν). This proves the claim about multiplicity. Theorem 4.7. Suppose λ 1, λ 2 P are dominant integral weights. There is an equivalence O λ1 Oλ2 that takes (w λ 1 ) to (w λ 2 ). Similarly, it takes (w λ 1 ) to (w λ 2 ) and L(w λ 1 ) to L(w λ 2 ). Proof. Firstly, we reduce to the case λ 1 λ 2 is dominant. This implies the general case, for it would imply that O λ1 and O λ2 are each equivalent to O λ1+λ 2. Now we apply proposition 4.6 with λ = λ 1 and µ = λ 2. If λ is the finite dimensional irreducible of highest weight λ 1 λ 2, we obtain functors ϕ = pr λ1 (V ι λ2 ( )) : O λ2 O λ1 and ϕ = pr λ2 (V ι λ1 ( )) : O λ1 O λ2. Recall that by Remark 4.2, ϕ and ϕ are mutually adjoint (on both sides). Note that proposition 4.6 implies that ϕ( (w λ 2 )) = (w λ 1 ). In fact, we also have ϕ ( (w λ 1 )) = (w λ 2 ). This is because λ 2 + ρ is strictly dominant, so Stab W (λ 2 + ρ) = {1}; thus w = w. In the notation of the proof of the proposition, this means that v i = λ µ (the highest weight, of which there is only one). Thus there is only one term in the filtration, (w λ 2 ) = (w λ 2 ). Hence, the module is isomorphic to (w λ 2 ). Now, let G = ϕ ϕ, an endofunctor of O λ1. Since ϕ and ϕ are adjoint, there is a adjunction unit: a 3

4 natural transformation η : Id G. We obtain a map η (w λ1) : (w λ 1 ) ϕ ϕ( (w λ 1 )). This map is nonzero for the following reason. Consider the following composition, which is required to be Id ϕ by adjunction properties: ϕ ϕ η ϕ ϕ ϕ ε ϕ ϕ Here ε is the adjunction counit. The first arrow is ϕ η, which must therefore be a monomorphism for each object M (it has a right inverse). In particular, ϕ(η (w λ1)) is injective, and hence nonzero. This means that η (w λ1) must be nonzero. Since End g ( (ν)) = C (for any ν h ), it follows that a nonzero map is an isomorphism; in particular ϕ ϕ defines an isomorphism on Verma modules. We now show that this implies we have an equivalence of categories. The functor G must be an isomorphism on any object that admits a standard filtration. This can be proved by induction on the length of the filtration (the base case being length 1), by considering a short exact sequence coming from the filtration. Suppose we have 0 F M (ν) 0 for some weight ν and module F with standard filtration. We obtain the following diagram by applying G: 0 F M (ν) 0 η F η M η (ν) 0 G(F ) G(M) G( (ν)) 0 The diagram commutes because η is a natural transformation. By induction we know that G is an isomorphism on the first and last terms, and it is exact. Therefore the five-lemma allows us to conclude that the middle map is an isomorphism; this completes the induction. Since projective objects admit standard filtrations (as O λ is a highest-weight category), G is an isomorphism on projective objects. It now follows that G is an isomorphism on all objects, by applying a similar argument to a truncated projective resolution of an arbitrary object (recall that O λ has enough projectives), P 1 P 0 M 0. This means that η defines a natural isomorphism between ϕ ϕ and the identity functor. The same proof works in the opposite direction (i.e. for ϕ ϕ ), as ϕ and ϕ are biadjoint. Thus ϕ and ϕ define an equivalence of categories. To check the behaviour on dual Verma modules, we simply use Proposition 4.3. The statement about behaviour on simples follows by taking maximal quotients of Verma modules (which are unique). Even in the case where λ 2 is not dominant, we can say some things. Theorem 4.8. Take λ 1, λ 2 to satisfy the conditions of Proposition 4.6 with λ = λ 1 and µ = λ 2. We have that pr µ (V L(w λ 1 )) is zero unless w is equal to the (unique) longest element in the coset wstab W (λ 2 +ρ), in which case it is L(w λ 2 ). Proof. We continue to write ϕ( ) = pr µ (V ι λ ( )). We choose a coset wstab W (λ 2 + ρ). Let u be the longest element of this coset, and assume w u. Then ϕ( (w λ 1 )) = (u λ 2 ) = ϕ( (u λ 1 )). We also have (by the results of the first lecture) that (u λ 1 ) (w λ 1 ). Let C be the cokernel of this inclusion, so that we have a short exact sequence: Applying the exact functor ϕ gives: 0 (u λ 1 ) (w λ 1 ) C 0 0 (u λ 2 ) (w λ 2 ) ϕ(c) 0 But the first two objects are the same (u λ 2 = w λ 2 ), and an injective endomorphism of a Verma module is an isomorphism. This means that the first map is an isomorphism (it is injective by exactness). This in turn forces ϕ(c) = 0. Hence, ϕ annihilates any subquotient of C, including L(w λ 2 ). 4

5 On the other hand, every ϕ( (w λ 1 )) = (w λ 2 ) so that every Verma module is in the image of ϕ. The classes of Verma modules span K 0 (O λ2 ), so the exact functor ϕ induces a surjection of Grothendieck groups [ϕ] : K 0 (O λ1 ) K 0 (O λ2 ). By rank considerations, it follows that ϕ(l(u λ 1 )) must be nonzero when u is the longest element in its coset; K 0 (O λ2 ) has rank equal to the number of such simples, and the images of the other simples are zero. It remains to determine ϕ(l(u λ 1 )). In view of Remark 4.5, the costandard objects (w λ 1 ) behave similarly to (w λ 1 ) under tensoring with V. Analogously to (u λ 2 ) ϕ(l(u λ 1 )), we have ϕ(l(u λ 1 )) (u λ 2 ). Composing these two maps, and using the fact from the previous lecture that dim(hom O ( (λ), (µ))) = δ λ,µ, we see that a (nonzero) map must be (a scalar multiple of) projection onto L(u λ 2 ) followed by inclusion into (u λ 2 ). It immediately follows that ϕ(l(u λ 1 )) = L(u λ 2 ). Corollary 4.9. Let λ + ρ be dominant, and w be the longest element in the coset wstab W (λ + ρ). We have the following equality of multiplicities: [ (w λ) : L(w λ)] = [ (w 0) : L(w 0)] Proof. This follows by exactness of ϕ and the characterisation of the action of ϕ on simple objects in the previous theorem (we take λ 1 = 0, λ 2 = λ). Remark The projection functors satisfy the following transitivity property. Suppose that λ 1, λ 2, λ 3 are all dominant weights such that λ 1 λ 2 and λ 2 λ 3 are dominant. Assume further that Stab W (λ 1 + ρ) Stab W (λ 2 + ρ) Stab W (λ 3 + ρ). For i, j {1, 2, 3} with i < j, let ϕ i,j : O λj O λi be the projection functor pr λi (V i,j ι λj ()), where V i,j is the finite-dimensional irreducible g-module of highest weight λ i λ j. Then we have ϕ 1,2 ϕ 2,3 = ϕ1,3. This is left as an exercise. 5. Projections to the Walls and Reflection Functors In this section we assume that λ 1 and λ 2 satisfy the conditions of Proposition 4.6 with λ = λ 1 and µ = λ 2, but we further require that there is a unique i such that α i, λ 2 + ρ = 0 (the inner products with other simple roots are positive). This means that λ 2 + ρ is on the wall of the dominant Weyl chamber associated to α i. Proposition 5.1. Let ϕ : O λ2 O λ1 and ϕ : O λ2 O λ1 be as in Proposition 4.6. Then we have ϕ ϕ : O λ1 O λ1. If w λ 1 ws i λ 1, we have the following short exact sequence: Otherwise we have: 0 (w λ 1 ) ϕ ϕ ( (w λ 1 )) (ws i λ 1 ) 0 0 (ws i λ 1 ) ϕ ϕ ( (w λ 1 )) (w λ 1 ) 0 Proof. This immediately follows from Proposition 4.6, noting that Stab W (λ 2 + ρ) = {1, s i }, and that weight spaces in the same Weyl group orbit have the same dimension (so the s i (λ 1 λ 2 ) weight space is one dimensional). Note that the Verma with lower weight is the quotient, and the Verma with larger weight is the submodule, because extensions can only exist in one direction. Definition 5.2. The functor ϕ ϕ is an endofunctor of O 0. We call this P i (the subscript i corresponds to the simple root α i ). Note that it is a self-adjoint exact functor. Proposition 5.3. The Grothendieck group of O 0 can be identified with ZW. The functor P i induces an endomorphism of K 0 (O 0 ) which is given by right multiplication by 1 + s i in ZW. Proof. We may take (w 0) to be a Z-basis of K 0 (O 0 ) (this follows from the highest-weight structure discussed in the previous lecture). This provides the identification with ZW (as abelian groups). An exact functor respect the relations of the Grothendieck group, and therefore defines an endomorphism of the Grothendieck group (as an abelian group). Proposition 5.1 implies that the image of the endomorphism associated to the functor P i evaluated on (w 0) is [ (w 0)] + [ (ws i 0)]. Under our identification, this becomes w w + ws i = w(1 + s i ). 5

6 6. Projective Objects in Category O Proposition 6.1. The object (0) is projective in the category O 0. Proof. This was in the preceding lecture (it is a consequence of the fact that 0 is maximal in W 0, so any vector of weight 0 is automatically a singular vector; this implies Hom O0 ( (0), ) is an exact functor). Definition 6.2. For w W, choose a reduced expression w = s i1 s i2 s il. Let P w = P il P i2 P i1 ( (0)). The following theorem was proved in the previous lecture, but we include a slightly different proof, which will be important later. Theorem 6.3. The category O 0 has enough projectives. The object P w is the projective cover of L(w 0) plus a direct sum of projective covers of L(w 0) with w smaller than w in the Bruhat order. Proof. Firstly, since P i is self-adjoint and exact, it takes projectives to projectives. This is because if P is a projective object, M Hom O0 (P, P i (M)) is the composition of two exact functors, hence exact. But adjointness makes this equal to M Hom O0 (P i (P ), M). The exactness of this functor is precisely the statement that P i (P ) is projective. It immediately follows that each P w is projective. By iterating Proposition 5.1, we find that P w is filtered by subquotients equal to Verma modules. Moreover, the Verma modules that appear can be described using the Proposition in the following way. Inductively, we expand (1 + s i1 )(1 + s i2 ) (1 + s il ) = w W m ww, the coefficients m w are precisely the multiplicities of the Verma modules (w 0) in this filtration. We observe that we have one instance of (w 0) (which is in fact a quotient of P w, as its highest weight is the lowest among modules appearing in the filtration). Note also that all other terms correspond to elements of W that can be written by removing simple reflections from a reduced expression for w, i.e. lower in the Bruhat order. If P is a projective module, dim(hom O (P, L(w 0))) is the multiplicity of the projective cover of L(w 0) in P. We use this as follows. Using the fact that in the Grothendieck group, [P (λ)] equals [ (λ)] plus higher order terms, the only possible projective covers arising in P w are P (µ), where µ = w 0, with w w. Since (w 0) occurs exactly once in the filtration, we have exactly one summand isomorphic to P (w 0). 6

### LECTURE 10: KAZHDAN-LUSZTIG BASIS AND CATEGORIES O

LECTURE 10: KAZHDAN-LUSZTIG BASIS AND CATEGORIES O IVAN LOSEV Introduction In this and the next lecture we will describe an entirely different application of Hecke algebras, now to the category O. In the

### 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,

### 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,

### Category O and its basic properties

Category O and its basic properties Daniil Kalinov 1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h b g. Define n = α>0 g α, n = α

### 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

### A relative version of Kostant s theorem

A relative version of Kostant s theorem 1 University of Vienna Faculty of Mathematics Srni, January 2015 1 supported by project P27072 N25 of the Austrian Science Fund (FWF) This talk reports on joint

### Representations of semisimple Lie algebras

Chapter 14 Representations of semisimple Lie algebras In this chapter we study a special type of representations of semisimple Lie algberas: the so called highest weight representations. In particular

### 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

### 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

### 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

### 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

### 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

### NOTES ON SPLITTING FIELDS

NOTES ON SPLITTING FIELDS CİHAN BAHRAN I will try to define the notion of a splitting field of an algebra over a field using my words, to understand it better. The sources I use are Peter Webb s and T.Y

### A pairing in homology and the category of linear complexes of tilting modules for a quasi-hereditary algebra

A pairing in homology and the category of linear complexes of tilting modules for a quasi-hereditary algebra Volodymyr Mazorchuk and Serge Ovsienko Abstract We show that there exists a natural non-degenerate

### 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

### LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F)

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) IVAN LOSEV In this lecture we will discuss the representation theory of the algebraic group SL 2 (F) and of the Lie algebra sl 2 (F), where F is

### ON AN INFINITE LIMIT OF BGG CATEGORIES O

ON AN INFINITE LIMIT OF BGG CATEGORIES O KEVIN COULEMBIER AND IVAN PENKOV Abstract. We study a version of the BGG category O for Dynkin Borel subalgebras of rootreductive Lie algebras, such as g = gl(

### Highest-weight Theory: Verma Modules

Highest-weight Theory: Verma Modules Math G4344, Spring 2012 We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or,

### KOSZUL DUALITY FOR STRATIFIED ALGEBRAS II. STANDARDLY STRATIFIED ALGEBRAS

KOSZUL DUALITY FOR STRATIFIED ALGEBRAS II. STANDARDLY STRATIFIED ALGEBRAS VOLODYMYR MAZORCHUK Abstract. We give a complete picture of the interaction between the Koszul and Ringel dualities for graded

### 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 :

### LECTURE 20: KAC-MOODY ALGEBRA ACTIONS ON CATEGORIES, II

LECTURE 20: KAC-MOODY ALGEBRA ACTIONS ON CATEGORIES, II IVAN LOSEV 1. Introduction 1.1. Recap. In the previous lecture we have considered the category C F := n 0 FS n -mod. We have equipped it with two

### MAT 445/ INTRODUCTION TO REPRESENTATION THEORY

MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations

### INTRODUCTION TO LIE ALGEBRAS. LECTURE 7.

INTRODUCTION TO LIE ALGEBRAS. LECTURE 7. 7. Killing form. Nilpotent Lie algebras 7.1. Killing form. 7.1.1. Let L be a Lie algebra over a field k and let ρ : L gl(v ) be a finite dimensional L-module. Define

### arxiv: v1 [math.rt] 11 Sep 2009

FACTORING TILTING MODULES FOR ALGEBRAIC GROUPS arxiv:0909.2239v1 [math.rt] 11 Sep 2009 S.R. DOTY Abstract. Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of

### Qualifying Exam Syllabus and Transcript

Qualifying Exam Syllabus and Transcript Qiaochu Yuan December 6, 2013 Committee: Martin Olsson (chair), David Nadler, Mariusz Wodzicki, Ori Ganor (outside member) Major Topic: Lie Algebras (Algebra) Basic

### 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.

### TCC Homological Algebra: Assignment #3 (Solutions)

TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate

### DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY C.M. RINGEL)

DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY CM RINGEL) C BOWMAN, SR DOTY, AND S MARTIN Abstract We give an algorithm for working out the indecomposable

### DUALITIES AND DERIVED EQUIVALENCES FOR CATEGORY O

DUALITIES AND DERIVED EQUIVALENCES FOR CATEGORY O KEVIN COULEMBIER AND VOLODYMYR MAZORCHUK Abstract. We determine the Ringel duals for all blocks in the parabolic versions of the BGG category O associated

### KOSZUL DUALITY FOR STRATIFIED ALGEBRAS I. QUASI-HEREDITARY ALGEBRAS

KOSZUL DUALITY FOR STRATIFIED ALGEBRAS I. QUASI-HEREDITARY ALGEBRAS VOLODYMYR MAZORCHUK Abstract. We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary

### H(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).

92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported

### THE THEOREM OF THE HIGHEST WEIGHT

THE THEOREM OF THE HIGHEST WEIGHT ANKE D. POHL Abstract. Incomplete notes of the talk in the IRTG Student Seminar 07.06.06. This is a draft version and thought for internal use only. The Theorem of the

### Lie Algebra Cohomology

Lie Algebra Cohomology Carsten Liese 1 Chain Complexes Definition 1.1. A chain complex (C, d) of R-modules is a family {C n } n Z of R-modules, together with R-modul maps d n : C n C n 1 such that d d

### Modular representation theory

Modular representation theory 1 Denitions for the study group Denition 1.1. Let A be a ring and let F A be the category of all left A-modules. The Grothendieck group of F A is the abelian group dened by

### is an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent

Lecture 4. G-Modules PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 4. The categories of G-modules, mostly for finite groups, and a recipe for finding every irreducible G-module of a

### A BRIEF REVIEW OF ABELIAN CATEGORIFICATIONS

Theory and Applications of Categories, Vol. 22, No. 19, 2009, pp. 479 508. A BRIEF REVIEW OF ABELIAN CATEGORIFICATIONS MIKHAIL KHOVANOV, VOLODYMYR MAZORCHUK CATHARINA STROPPEL Abstract. This article contains

### LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated

### MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to

### 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

### D-MODULES: AN INTRODUCTION

D-MODULES: AN INTRODUCTION ANNA ROMANOVA 1. overview D-modules are a useful tool in both representation theory and algebraic geometry. In this talk, I will motivate the study of D-modules by describing

### 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.

### ON THE MAXIMAL PRIMITIVE IDEAL CORRESPONDING TO THE MODEL NILPOTENT ORBIT

ON THE MAXIMAL PRIMITIVE IDEAL CORRESPONDING TO THE MODEL NILPOTENT ORBIT HUNG YEAN LOKE AND GORDAN SAVIN Abstract. Let g = k s be a Cartan decomposition of a simple complex Lie algebra corresponding to

### Braid group actions on categories of coherent sheaves

Braid group actions on categories of coherent sheaves MIT-Northeastern Rep Theory Seminar In this talk we will construct, following the recent paper [BR] by Bezrukavnikov and Riche, actions of certain

### 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

### THE COMPOSITION SERIES OF MODULES INDUCED FROM WHITTAKER MODULES

THE COMPOSITION SERIES OF MODULES INDUCED FROM WHITTAKER MODULES DRAGAN MILIČIĆ AND WOLFGANG SOERGEL Abstract. We study a category of representations over a semisimple Lie algebra, which contains category

### 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

### ON DEGENERATION OF THE SPECTRAL SEQUENCE FOR THE COMPOSITION OF ZUCKERMAN FUNCTORS

ON DEGENERATION OF THE SPECTRAL SEQUENCE FOR THE COMPOSITION OF ZUCKERMAN FUNCTORS Dragan Miličić and Pavle Pandžić Department of Mathematics, University of Utah Department of Mathematics, Massachusetts

### 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

### WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES

WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,

### 7 Rings with Semisimple Generators.

7 Rings with Semisimple Generators. It is now quite easy to use Morita to obtain the classical Wedderburn and Artin-Wedderburn characterizations of simple Artinian and semisimple rings. We begin by reminding

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

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture I Set-up. Let K be an algebraically closed field. By convention all our algebraic groups will be linear algebraic

### REPRESENTATION THEORY. WEEK 4

REPRESENTATION THEORY. WEEK 4 VERA SERANOVA 1. uced modules Let B A be rings and M be a B-module. Then one can construct induced module A B M = A B M as the quotient of a free abelian group with generators

### On certain family of B-modules

On certain family of B-modules Piotr Pragacz (IM PAN, Warszawa) joint with Witold Kraśkiewicz with results of Masaki Watanabe Issai Schur s dissertation (Berlin, 1901): classification of irreducible polynomial

### ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS

ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS Your Name: Conventions: all rings and algebras are assumed to be unital. Part I. True or false? If true provide a brief explanation, if false provide a counterexample

### NOTES ON RESTRICTED INVERSE LIMITS OF CATEGORIES

NOTES ON RESTRICTED INVERSE LIMITS OF CATEGORIES INNA ENTOVA AIZENBUD Abstract. We describe the framework for the notion of a restricted inverse limit of categories, with the main motivating example being

### AN UPPER BOUND FOR SIGNATURES OF IRREDUCIBLE, SELF-DUAL gl(n, C)-REPRESENTATIONS

AN UPPER BOUND FOR SIGNATURES OF IRREDUCIBLE, SELF-DUAL gl(n, C)-REPRESENTATIONS CHRISTOPHER XU UROP+ Summer 2018 Mentor: Daniil Kalinov Project suggested by David Vogan Abstract. For every irreducible,

### 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

### 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

### The Cartan Decomposition of a Complex Semisimple Lie Algebra

The Cartan Decomposition of a Complex Semisimple Lie Algebra Shawn Baland University of Colorado, Boulder November 29, 2007 Definition Let k be a field. A k-algebra is a k-vector space A equipped with

### SCHUR-WEYL DUALITY FOR U(n)

SCHUR-WEYL DUALITY FOR U(n) EVAN JENKINS Abstract. These are notes from a lecture given in MATH 26700, Introduction to Representation Theory of Finite Groups, at the University of Chicago in December 2009.

### Primitive Ideals and Unitarity

Primitive Ideals and Unitarity Dan Barbasch June 2006 1 The Unitary Dual NOTATION. G is the rational points over F = R or a p-adic field, of a linear connected reductive group. A representation (π, H)

### 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

### On some combinatorial aspects of Representation Theory

On some combinatorial aspects of Representation Theory Doctoral Defense Waldeck Schützer schutzer@math.rutgers.edu Rutgers University March 24, 2004 Representation Theory and Combinatorics p.1/46 Overview

### The real root modules for some quivers.

SS 2006 Selected Topics CMR The real root modules for some quivers Claus Michael Ringel Let Q be a finite quiver with veretx set I and let Λ = kq be its path algebra The quivers we are interested in will

### Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups

Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Shrawan Kumar Talk given at AMS Sectional meeting held at Davidson College, March 2007 1 Hermitian eigenvalue

### Equivariant Algebraic K-Theory

Equivariant Algebraic K-Theory Ryan Mickler E-mail: mickler.r@husky.neu.edu Abstract: Notes from lectures given during the MIT/NEU Graduate Seminar on Nakajima Quiver Varieties, Spring 2015 Contents 1

### 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

### A Koszul category of representations of finitary Lie algebras

A Koszul category of representations of finitary Lie algebras Elizabeth Dan-Cohen a,1,, Ivan Penkov a,2, Vera Serganova b,3 a Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany b University

### The Kac-Wakimoto Character Formula

The Kac-Wakimoto Character Formula The complete & detailed proof. Jean Auger September-October 2015 The goal of this document is to present an intelligible proof of the character formula that goes by the

### 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

### 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

### QUIVERS AND LATTICES.

QUIVERS AND LATTICES. KEVIN MCGERTY We will discuss two classification results in quite different areas which turn out to have the same answer. This note is an slightly expanded version of the talk given

### Vanishing Ranges for the Cohomology of Finite Groups of Lie Type

for the Cohomology of Finite Groups of Lie Type University of Wisconsin-Stout, University of Georgia, University of South Alabama The Problem k - algebraically closed field of characteristic p > 0 G -

### 16.2. Definition. Let N be the set of all nilpotent elements in g. Define N

74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the

### A PROOF OF BOREL-WEIL-BOTT THEOREM

A PROOF OF BOREL-WEIL-BOTT THEOREM MAN SHUN JOHN MA 1. Introduction In this short note, we prove the Borel-Weil-Bott theorem. Let g be a complex semisimple Lie algebra. One basic question in representation

### 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

### 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

### Conformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G.

Conformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G. Giorgia Fortuna May 04, 2010 1 Conformal blocks for a chiral algebra. Recall that in Andrei s talk [4], we studied what it means

### ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need

### 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

### Critical level representations of affine Kac Moody algebras

Critical level representations of affine Kac Moody algebras Peter Fiebig Emmy Noether Zentrum Universität Erlangen Nürnberg Isle of Skye May 2010 Affine Kac Moody algebras Let g be a complex simple Lie

### 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

### MULTIPLICITY FORMULAS FOR PERVERSE COHERENT SHEAVES ON THE NILPOTENT CONE

MULTIPLICITY FORMULAS FOR PERVERSE COHERENT SHEAVES ON THE NILPOTENT CONE A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial

### Exercises on chapter 0

Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that

### Lecture 5: Admissible Representations in the Atlas Framework

Lecture 5: Admissible Representations in the Atlas Framework B. Binegar Department of Mathematics Oklahoma State University Stillwater, OK 74078, USA Nankai Summer School in Representation Theory and Harmonic

### 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

### 4.1. Paths. For definitions see section 2.1 (In particular: path; head, tail, length of a path; concatenation;

4 The path algebra of a quiver 41 Paths For definitions see section 21 (In particular: path; head, tail, length of a path; concatenation; oriented cycle) Lemma Let Q be a quiver If there is a path of length

### PERVERSE SHEAVES. Contents

PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a

### 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

### De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)

II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the

### Lemma 1.3. The element [X, X] is nonzero.

Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

### arxiv: v1 [math.rt] 31 Oct 2008

Cartan Helgason theorem, Poisson transform, and Furstenberg Satake compactifications arxiv:0811.0029v1 [math.rt] 31 Oct 2008 Adam Korányi* Abstract The connections between the objects mentioned in the

### DEFORMATIONS OF ALGEBRAS IN NONCOMMUTATIVE ALGEBRAIC GEOMETRY EXERCISE SHEET 1

DEFORMATIONS OF ALGEBRAS IN NONCOMMUTATIVE ALGEBRAIC GEOMETRY EXERCISE SHEET 1 TRAVIS SCHEDLER Note: it is possible that the numbers referring to the notes here (e.g., Exercise 1.9, etc.,) could change

### IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS. Contents

IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS NEEL PATEL Abstract. The goal of this paper is to study the irreducible representations of semisimple Lie algebras. We will begin by considering two

### arxiv: v1 [math.rt] 14 Nov 2007

arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof

### SCHUR-WEYL DUALITY FOR QUANTUM GROUPS

SCHUR-WEYL DUALITY FOR QUANTUM GROUPS YI SUN Abstract. These are notes for a talk in the MIT-Northeastern Fall 2014 Graduate seminar on Hecke algebras and affine Hecke algebras. We formulate and sketch

### AHAHA: Preliminary results on p-adic groups and their representations.

AHAHA: Preliminary results on p-adic groups and their representations. Nate Harman September 16, 2014 1 Introduction and motivation Let k be a locally compact non-discrete field with non-archimedean valuation

### Semistable Representations of Quivers

Semistable Representations of Quivers Ana Bălibanu Let Q be a finite quiver with no oriented cycles, I its set of vertices, k an algebraically closed field, and Mod k Q the category of finite-dimensional