Relaxed Highest Weight Modules from D-Modules on the Kashiwara Flag Scheme. Claude Eicher, ETH Zurich November 29, PDF Free Download

Reaxed Highest Weight Modues from D-Modues on the Kashiwara Fag Scheme Caude Eicher, ETH Zurich November 29, 2016 1

Reaxed highest weight modues for ŝ 2 after Feigin, Semikhatov, Sirota,Tipunin Introduction to ocaization on the affine fag variety Setup Overview of resuts 2

Reaxed highest weight modues for ŝ 2 after Feigin, Semikhatov, Sirota,Tipunin

We start with the Lie agebra s 2 = C e C h C f and define ŝ 2 = s 2 C C[z, z 1 ] C K, where K is centra and [X z m, Y z n ] = [X, Y ] z m+n + mδ m+n,0 Tr(XY )K. This endows ŝ 2 with the structure of a Lie agebra. Up to the derivation eement this ( defines) the affine Kac-Moody 2 2 agebra with Cartan matrix. 2 2 3

Definition (Semikhatov-Sirota 97) Let µ 1, µ 2 C and t C \{0}. The reaxed Verma modue R µ1,µ 2,t is the ŝ 2-modue generated from a vector v that satisfies the annihiation conditions (e z n )v = (h z n )v = (f z n )v = 0 n 1 and the reations (f 1)(e 1)v = µ 1 µ 2 v (h 1)v = (1 + µ 1 + µ 2 )v Kv = (t 2)v by a free action of e z n, f z n, h z n, n 1. 4

Different µ 1, µ 2 can give rise to isomorphic R µ1,µ 2,t and it is easy to write out the condition. 5

Different µ 1, µ 2 can give rise to isomorphic R µ1,µ 2,t and it is easy to write out the condition. To get a first impression about the structure of R µ1,µ 2,t we can ook at the s 2 C 1-submodue generated by v. It is a weight modue with weights (1 + µ 1 + µ 2 ) + 2 Z, each of which has mutipicity one. We have (f 1)(e 1) µ j +1 v = 0 and (e 1)(f 1) µ j v = 0 if these expressions are actuay defined and µ j 0 in the second case. 5

µ 1 / Z, µ 2 / Z (µ 1 + µ 2 + 1) v µ 1 Z 0, µ 2 / Z µ 1 µ 2 + 1 v µ 1 Z <0, µ 2 / Z µ 1 µ 2 1 v 6

µ 1 / Z, µ 2 / Z case (0) (µ 1 + µ 2 + 1) v µ 1 Z 0, µ 2 / Z case (1, +) µ 1 µ 2 + 1 v µ 1 Z <0, µ 2 / Z case (1, ) µ 1 µ 2 1 v 6

µ 1 Z <0, µ 2 Z 0 µ 1 µ 2 1 µ 2 µ 1 + 1 µ 1 Z 0, µ 2 Z 0, µ 1 µ 2 v µ 2 µ 1 + 1 µ 1 µ 2 + 1 v µ 1 Z <0, µ 2 Z <0, µ 1 µ 2 µ 1 µ 2 1 µ 2 µ 1 1 v 7

µ 1 Z <0, µ 2 Z 0 case (2, +) µ 1 µ 2 1 µ 2 µ 1 + 1 µ 1 Z 0, µ 2 Z 0, µ 1 µ 2 case (2, ++) v µ 2 µ 1 + 1 µ 1 µ 2 + 1 v µ 1 Z <0, µ 2 Z <0, µ 1 µ 2 case (2, ) µ 1 µ 2 1 µ 2 µ 1 1 v 7

Coming back to R µ1,µ 2,t, we note that (e 1)(f 1) µ 1 v = 0 impies that the submodue generated by (f 1) µ 1 v is isomorphic to a Verma modue of highest weight λ = µ 1 µ 2 1. We wi denote it by M λ,t. 8

Consider the automorphism of ŝ 2 sending K K and e z n e z n+θ f z n f z n θ h z n h z n + θδ n,0 K. 9

Consider the automorphism of ŝ 2 sending K K and e z n e z n+θ f z n f z n θ h z n h z n + θδ n,0 K. The vector w = (e 1) µ 1+1 v satisfies (e z 1 )w = (h z 1 )w = (f z 0 )w = 0 (h 1 + (t 2))w = (t + µ 1 µ 2 1)w. 9

Consider the automorphism of ŝ 2 sending K K and e z n e z n+θ f z n f z n θ h z n h z n + θδ n,0 K. The vector w = (e 1) µ 1+1 v satisfies (e z 1 )w = (h z 1 )w = (f z 0 )w = 0 (h 1 + (t 2))w = (t + µ 1 µ 2 1)w. Thus w generates a submodue of R µ1,µ 2,t which is isomorphic to a Verma modue twisted by the automorphism for θ = 1. We wi denote it by M (1) t+µ 1 µ 2 1,t. 9

So we have the foowing embeddings (1, +) R µ1,µ 2,t M (1) t+µ 1 µ 2 1,t (1, ) M µ1 µ 2 1,t R µ1,µ 2,t (2, +) M µ1 µ 2 1,t R µ1,µ 2,t M (1) t+µ 2 µ 1 1,t (2, ++) R µ1,µ 2,t M (1) t+µ 2 µ 1 1,t M(1) t+µ 1 µ 2 1,t (2, ) M µ1 µ 2 1,t M µ2 µ 1 1,t R µ1,µ 2,t The goa of Semikhatov-Sirota 97 is to describe which modues M λ,t, M (1) λ,t or R µ 1,µ 2,t embed into R µ1,µ 2,t. 10

Simpicity of R µ1,µ 2,t From the above we concude that µ 1 / Z and µ 2 / Z is a necessary condition for R µ1,µ 2,t to be simpe. 11

Simpicity of R µ1,µ 2,t From the above we concude that µ 1 / Z and µ 2 / Z is a necessary condition for R µ1,µ 2,t to be simpe. Theorem (Semikhatov-Sirota 97) R µ1,µ 2,t simpe µ 1 / Z and µ 2 / Z and r, s Z >0 µ 1 µ 2 = r st or µ 2 µ 1 = r st 11

The description of the so-caed embedding diagrams for R µ1,µ 2,t is the main resut of Semikhatov-Sirota 97. These diagrams are abeed by combining I, II, III ± determined by the row (0), (1, +), (1, ), (2, ),... determined by the coumn. 12

Exampe: case III 0 +(2, +) 13

Introduction to ocaization on the affine fag variety

Let g be a finite dimensiona semisimpe Lie agebra over C. The ceebrated theorem of Beiinson-Bernstein 81 states that the functor of goba sections is an exact equivaence of categories between the D-modues on the fag variety twisted by the ine bunde associated to a reguar dominant weight and the g-modues of the corresponding centra character. At present the fu anaogue of this statement in the case of affine Kac-Moody agebras is not known. Postponing definitions, et us start by pointing out reated theorems in the case of affine Kac-Moody agebras g. 14

A first possibiity, in case g is untwisted, is to consider the thin fag variety defined as a quotient of the oop group by the Iwahori group scheme X thin = L G/ L + I (Beiinson-Drinfed, Pappas-Rapoport 08 and others). Here G is a semisimpe agebraic group. Beiinson-Drinfed define a category of twisted right D-modues on X thin and a functor of goba sections Γ(X thin, ) anding in g mod. Theorem (Beiinson-Drinfed, P. Shan 11) Let λ + ρ be reguar antidominant. The functor Γ(X thin, ) between the λ-twisted right D-modues on X thin and g mod is exact and faithfu. 15

A second possibiity is to consider the Kashiwara fag scheme X. As we wi detai ater, it is a scheme, not ocay of finite type, but having an open cover by affine spaces of countabe dimension. The finite dimensiona Schubert ces X w can be defined as subschemes of X and one again has a notion of twisted D-modues on X and a functor of goba sections. Reca the notion of the Verma modue M(µ) = U g U b C µ of highest weight µ h. 16

The ast two theorems can be combined into Theorem (Frenke-Gaitsgory 09) Let λ + ρ be reguar antidominant. Γ(X thin, ) defines an exact equivaence between the category of λ-twisted right D-modues on X thin that are equivariant for the pro-unipotent radica of L + I and the bock of category O of g defined by λ. 17

Setup

Affine Kac-Moody agebras (h, (α i ) i I h, (h i ) i I h) affine Kac-Moody data 18

Affine Kac-Moody agebras (h, (α i ) i I h, (h i ) i I h) affine Kac-Moody data g affine Kac-Moody (Lie) agebra 18

Affine Kac-Moody agebras (h, (α i ) i I h, (h i ) i I h) affine Kac-Moody data g affine Kac-Moody (Lie) agebra g = n h n trianguar decomposition into positive and negative part n and n and the Cartan subagebra h b ( ) = n ( ) h Bore and opposite Bore subagebra 18

n Lie idea of n given by the root spaces associated to the negative roots of height, Z 0, simiary n 19

n Lie idea of n given by the root spaces associated to the negative roots of height, Z 0, simiary n g i = C f i h C e i, i I 19

n Lie idea of n given by the root spaces associated to the negative roots of height, Z 0, simiary n g i = C f i h C e i, i I p i = C e i b and n i p i = g, i I 19

n Lie idea of n given by the root spaces associated to the negative roots of height, Z 0, simiary n g i = C f i h C e i, i I p i = C e i b and n i p i = g, i I p i = C f i b and n i p i = g, i I 19

n Lie idea of n given by the root spaces associated to the negative roots of height, Z 0, simiary n g i = C f i h C e i, i I p i = C e i b and n i p i = g, i I p i = C f i b and n i p i = g, i I W Wey group of g. It acts ineary on h. For i I there is a simpe refection s i W. ρ h such that ρ(h i ) = 1 for a i I 19

Group schemes Pro-unipotent group scheme U = im exp(n/n ) 20

Group schemes Pro-unipotent group scheme U = im exp(n/n ) Pro-unipotent group scheme U(Ψ) U, where Ψ Φ >0 satisfies (Ψ + Ψ) Φ >0 Ψ, simiary U (Ψ) U. 20

Group schemes Pro-unipotent group scheme U = im exp(n/n ) Pro-unipotent group scheme U(Ψ) U, where Ψ Φ >0 satisfies (Ψ + Ψ) Φ >0 Ψ, simiary U (Ψ) U. Pro-unipotent group scheme U = U (Φ <0 ), where Φ <0 Φ <0 is the subset of negative roots of height. T = Spec C[P] = G m dim h agebraic torus B ( ) = T U ( ) Bore and opposite Bore group scheme 20

Kashiwara fag scheme Out of the affine Kac-Moody data and the attice P Kashiwara 90 constructs a scheme G with a distinguished point 1 G and commuting eft and right actions of P i and P i respectivey, for a i I. The Kashiwara fag scheme is defined as X = G/B (quotient by a ocay free action). The so-caed Tits extension W of W acts on G and X. On T -invariant subsets of X, the action of W factors through W. 21

Basic properties U 1B = A is an open subscheme of X caed big ce. 22

Basic properties U 1B = A is an open subscheme of X caed big ce. X = w W N(X w ), where N(X w ) = wu 1B 22

Basic properties U 1B = A is an open subscheme of X caed big ce. X = w W N(X w ), where N(X w ) = wu 1B X = w W X w, where X w = v w N(X v ) is B -invariant and quasi-compact and is the Bruhat partia order 22

For fixed w and a Z >0 arge enough U acts ocay freey on X w. The quotient X w = U \X w is a smooth quasi-projective variety (Shan-Varagnoo-Vasserot 14). 23

For fixed w and a Z >0 arge enough U acts ocay freey on X w. The quotient X w = U \X w is a smooth quasi-projective variety (Shan-Varagnoo-Vasserot 14). We have for 1 2 arge enough a commutative diagram X w 1 p 1 2 X w X w 2 The fibers of the projection p 1 2 embeddings. are affine spaces and are cosed 23

Twisted D-modues on X Ho(D X w (λ), X w ) Category of hoonomic right D-modues on (λ), λ P, with support in X w X w twisted by O X w 24

Twisted D-modues on X Ho(D X w (λ), X w ) Category of hoonomic right D-modues on (λ), λ P, with support in X w X w twisted by O X w It is an abeian category and every object has finite ength. 24

Twisted D-modues on X Ho(D X w (λ), X w ) Category of hoonomic right D-modues on (λ), λ P, with support in X w X w twisted by O X w It is an abeian category and every object has finite ength. It has a contravariant exact auto-equivaence D, the hoonomic duaity. 24

( ) Ho(λ, X w, X w, 0 ) M = (M ) 0, (γ 1 2 ) 1 2 0 Ho(D X w (λ), X w ) M γ 1 2 : p 1 2 M 1 = M2 γ 1 3 = γ 2 3 p 2 3 γ 1 2, 1 2 3 0 25

( ) Ho(λ, X w, X w, 0 ) M = (M ) 0, (γ 1 2 ) 1 2 0 Ho(D X w (λ), X w ) M γ 1 2 : p 1 2 M 1 = M2 γ 1 3 = γ 2 3 p 2 3 γ 1 2, 1 2 3 0 For any 0, M M is an exact equivaence. 25

( ) Ho(λ, X w, X w, 0 ) M = (M ) 0, (γ 1 2 ) 1 2 0 Ho(D X w (λ), X w ) M γ 1 2 : p 1 2 M 1 = M2 γ 1 3 = γ 2 3 p 2 3 γ 1 2, 1 2 3 0 For any 0, M M is an exact equivaence. Taking imits we get rid of the auxiiary choices X w, X w and 0 and define the category Ho(λ) of λ-twisted hoonomic right D-modues on X. 25

Cohomoogy groups For M Ho(λ) and j Z 0 define H j (X, M) = im H j (X w, M ), where H j (X w, M ) are the sheaf cohomoogy groups. 26

Cohomoogy groups For M Ho(λ) and j Z 0 define H j (X, M) = im H j (X w, M ), where H j (X w, M ) are the sheaf cohomoogy groups. If v g there is a m Z 0 such that [v, n +m ] n for a. Then v defines a C-inear map H j (X w +m, M +m) H j (X w, M ). In this way H j (X, M) becomes a g-modue. Define H j (X, M) = µ h H j (X, M) µ, where ( ) µ denotes the generaized weight space associated to µ. This is a g-submodue of H j (X, M). 26

Coordinates on N(X w ) For w W abbreviate Uw = U (Φ <0 wφ <0 ) U U w = U(Φ >0 wφ <0 ) U. The map (u 1, u 2 ) u 1 u 2 w1b defines a T -equivariant isomorphism of schemes Uw = U w N(Xw ). The image of 1 U w is the (finite dimensiona) Schubert ce X w in X. 27

Let w be such that s i w < w. Lemma We have X w s i X w = s i X w \ X si w = X w \ s i X si w. In the above coordinates on N(X w ) and s i N(X w ) = N(X si w ) the identity map s i X w \ X si w X w \ s i X si w is the isomorphism (U ( α i ) \ 1) U si w (U(α i ) \ 1) s i U si w (e zf i, h i (z) 1 uh i (z)) (e z 1 e i, s 1ũ(z) i s i ). Here z G m and s i = e e i e f i e e i. h i is considered as a group homomorphism G m T. Given u and z, ũ(z) U si w is uniquey determined by the condition e ze i u ũ(z)u(φ >0 s i wφ >0 ). 28

For s i w < w consider the ocay cosed affine embedding i w, : X w s i X w X w. 29

For s i w < w consider the ocay cosed affine embedding i w, : X w s i X w X w. Definition Define the right D X w (λ)-modue for λ P and α C (( ) R?w (λ, α) = i w,? Ω (α) C Ωs i Usi w ) iw, O X w (λ)? {,!}. 29

For s i w < w consider the ocay cosed affine embedding i w, : X w s i X w X w. Definition Define the right D X w (λ)-modue for λ P and α C (( ) R?w (λ, α) = i w,? Ω (α) C Ωs i Usi w ) iw, O X w (λ)? {,!}. Here we introduced the right D C -modue Ω (α) C = D C /(x x α)d C. The coordinate x on C is the one of U(α i ). Thus x = corresponds to X si w. 29

Before describing the cohomoogy of these D-modues, et us pause briefy and expain that X w s i X w can be understood as orbits for the subgroup L + I s i L + I of the Iwahori group L + I acting on X thin. Indeed, for s i w > w the L + I -orbit X w is aso a L + I s i L + I -orbit, as is s i X w. For s i w < w the L + I -orbit X w spits into two L + I s i L + I -orbits X w = (X w s i X w ) s i X si w. 30

The case of ŝ 2 X 1 X s0 X s0 s 1 X s0 s 1 s 0 X s1 s 1 X s1 X s1 s 0 s 1 X s1 s 0 X s1 s 0 s 1 s 1 X s1 s 0 s 1... s 1 X 1 s 1 X s0 s 1 X s0 s 1 s 1 X s0 s 1 s 0 The arrows indicate the cosure reations. 31

Overview of resuts

Pan We wi identify the goba sections of R?w (λ, α) as a g-modue for some choices of the parameters?, w, λ, α foowing the methods of Kashiwara-Tanisaki 95. We thereby extend severa of their resuts to a cass of non-highest weight representations which we ca reaxed highest weight because they generaize the ŝ 2 -representations that we have seen earier. We wi start by discussing the h-modue structure. 32

h-modue structure of goba sections Theorem We have isomorphisms of h-modues 1. H 0 (X w, R w (λ, α) ) = C[z, z 1 ] C S(n i /n ) C C si w λ+αα i 2. H 0 (X, R w (λ, α)) = C[z, z 1 ] C S n i C C si w λ+αα i Here z has weight α i. 33

Sketch of proof Consider the foowing factorization of i w, : X w s i X w X w X w s i X w X w N(X w ) = (U \U w ) U w X w. 34

Sketch of proof Consider the foowing factorization of i w, : X w s i X w X w X w s i X w X w N(X w ) = (U \U w ) U w X w. The first and third embedding are open and affine, whie the second embedding is cosed. By definition H 0 (X w, R w (λ, α) ) = H 0 (N(X w ), R w (λ, α) ) and there is the expicit description of the -direct image w.r.t. the second embedding κ w, : X w N(X w ) κ w, M = κ w, M C C[ 1,..., r ], where M is any right D-modue on X w and r = dim U \U w. 34

Cohomoogy vanishing Lemma H j (X w, R w (λ, α) ) = 0 and consequenty H j (X, R w (λ, α)) = 0 for j > 0. 35

Cohomoogy vanishing Lemma H j (X w, R w (λ, α) ) = 0 and consequenty H j (X, R w (λ, α)) = 0 for j > 0. This is again proven using the fact that H j (X w, R w (λ, α) ) = H j (N(X w ), R w (λ, α) ) and the above expicit description of the -direct image. 35

Definition Define the s 2 -modue for Λ, α C R s 2 (Λ, α) = U s 2 /(h + 2α + Λ, ef + (α + Λ)(α + 1)). 36

Definition Define the s 2 -modue for Λ, α C R s 2 (Λ, α) = U s 2 /(h + 2α + Λ, ef + (α + Λ)(α + 1)). When Λ Z 2, α Z and 1 Λ α 1 this is a singe isomorphism cass denoted by R s 2 (Λ) (case (2, +)). Its weight diagram is Λ Λ + 2 Λ 2 Λ 36

Reaxed Verma modues The generaization of R µ1,µ 2,t for arbitrary g is Definition Define the g-modue for λ P, α C ) R(λ, α) = U g U pi (C λ C R s 2 (λ(h i ), α). Here p i acts on C λ C R s 2 (λ(h i ), α) via the projection p i p i /n i = g i = {h h α i (h) = 0} g i. 37

The first observation one can make about this definition is that the underying h-modue of R(w λ, α) coincides with the h-modue H 0 (X, R w (λ, α)) described earier. 38

The first observation one can make about this definition is that the underying h-modue of R(w λ, α) coincides with the h-modue H 0 (X, R w (λ, α)) described earier. In the rest of the presentation we wi expain the cases in which we can prove that this induced modue is indeed the (dua of the) g-modue of goba sections. 38

The case w = s i Let M be a hoonomic right D-modue on X si = P 1 twisted by O P 1( λ(h i )). Let i s i, : X si X s i be the cosed embedding. Then i s i M = (i s i, M) 0 Ho(λ). 39

The case w = s i Let M be a hoonomic right D-modue on X si = P 1 twisted by O P 1( λ(h i )). Let i s i, : X si X s i be the cosed embedding. Then i s i M = (i s i, M) 0 Ho(λ). Theorem H j (X, i s i M) = U g U pi H j (P 1, M) as g-modue Together with the description of the cohomoogy of twisted D-modues obtained as direct images from X si s i X si = C P 1 (arxiv:1509.05299 [math.rt]) this gives a description of the g-modues H j (X, R?si (λ, α)) for j {0, 1} and a vaues of?, λ, α in terms of the above R(λ, α) and obvious modifications thereof. 39

Exact auto-equivaence s i of Ho(λ) The automorphism s := s i = e e i e f i e e i morphisms s + 4. : X w + X w of X descends to affine for w such that s i w < w and 40

Exact auto-equivaence s i of Ho(λ) The automorphism s := s i = e e i e f i e e i morphisms s + 4.The functor s + Ho ( D (s + X w + : X w + X w ) O w (λ) X of X descends to affine for w such that s i w < w and is an exact equivaence, X w ) Ho ( D O w (λ) X X w, X w ) Identifying (s + ) O X w (λ) = O X w (λ) we get an exact + auto-equivaence of Ho(λ).. 40

The case of R w (λ) Let us abbreviate the isomorphism cass R w (λ) = R w (λ, α) when α Z (trivia monodromy). 41

The case of R w (λ) Let us abbreviate the isomorphism cass R w (λ) = R w (λ, α) when α Z (trivia monodromy). Theorem Let λ + ρ be reguar antidominant. Then H 0 (X, R w (λ)) = R(w λ) as g-modue. 41

Sketch of proof Lemma We have an isomorphism of g i -modues H 0 (X, R w (λ)) µ = C w λ R s 2 ((w λ)(h i )) µ Z α i +w λ 42

Sketch of proof Lemma We have an isomorphism of g i -modues H 0 (X, R w (λ)) µ = C w λ R s 2 ((w λ)(h i )) µ Z α i +w λ Thus we have an induced morphism φ : R(w λ) H 0 (X, R w (λ)) of g-modues. Source and target coincide as h-modues. In order to prove that φ is an isomorphism it suffices to prove that it injects. 42

We have a short exact sequence 0 s i B w (λ) R w (λ) B si w (λ) 0 in Ho(λ). Here B w (λ) is the -direct image from the Schubert ce X w. 43

We have a short exact sequence 0 s i B w (λ) R w (λ) B si w (λ) 0 in Ho(λ). Here B w (λ) is the -direct image from the Schubert ce X w. We have H 1 (X, s i B w (λ)) = H 1 (X, B (λ)) s i w = 0. We get a surjection H 0 (X, R w (λ)) H 0 (X, B si w (λ)) and hence an injection ψ : H 0 (X, B si w (λ)) H 0 (X, R w (λ)). 43

Lemma R(w λ) does not have nonzero g i-finite vectors. 44

Lemma R(w λ) does not have nonzero g i-finite vectors. This emma impies Proposition Any nonzero submodue of R(w λ) intersects the submodue M(s i w λ) M(s i w λ) s i nontriviay. 44

Lemma R(w λ) does not have nonzero g i-finite vectors. This emma impies Proposition Any nonzero submodue of R(w λ) intersects the submodue M(s i w λ) M(s i w λ) s i nontriviay. Appy the proposition to ker φ. Note that φ M(s i w λ) is a nonzero mutipe of ψ and simiary for ψ s i to concude ker φ = 0. 44

Relaxed Highest Weight Modules from D-Modules on the Kashiwara Flag Scheme. Claude Eicher, ETH Zurich November 29, 2016