Equivariant compactifications of reductive groups Dmitri A. - PowerPoint PPT Presentation

Equivariant compactifications of reductive groups Dmitri A. Timashev Moscow State University

Abstract. We study equivariant projective compactifications of reductive groups obtained by closing the image of a group in the space of operators of a projective representation. (This is a “non-commutative generalization” of projective toric varieties.) We describe the structure and the mutual position of their orbits under the action of the doubled group by left/right multiplica- tions, the local structure in a neighborhood of a closed orbit, and obtain some conditions of normality and smoothness of a compactification. Our approach uses the theory of equivariant embeddings of spherical homogeneous spaces and of reductive algebraic semigroups. 1

1. Projective embeddings of reductive groups Let G be a connected reductive complex algebraic group. Examples. G = GL n ( C ), SL n ( C ), SO n ( C ), Sp n ( C ), ( C × ) n . Let G � P ( V ) be a faithful projective representation. It comes from a faithful rational linear representation � G � V , where � G → G is a finite cover. � → End V ⇒ G ֒ → P (End V ) G ֒ = Objective. Describe X = G ⊆ P (End V ) . Example 1. G = T = ( C × ) n an algebraic torus; λ 1 , . . . , λ m ∈ Z n the eigenweights of T = � ⇒ X is a projective toric variety T � V = corresponding to the polytope P = conv { λ 1 , . . . , λ m } . 2

Problem 1. Describe ( G × G ) -orbits in X : dimensions, represen- tatives, stabilizers, partial order by inclusion of closures. Problem 2. Describe the local structure of X . Problem 3. Normality of X . Problem 4. Smoothness of X . Relevant research. 1) Affine embeddings of reductive groups = reductive algebraic semigroups (Putcha–Renner, Vinberg [Vi95], Rittatore [Ri98]). 2) Regular group compactifications: cohomology (De Concini– Procesi [CP86], Strickland [St91]), cellular decomposition (Brion– Polo [BP00]). 3) Reductive varieties (Alexeev–Brion [AB04], [AB04’]). 3

Notation. G ) a maximal torus; Λ ≃ Z n the weight lattice, T ⊆ G (resp. � T ⊆ � T → C × , � weights ∀ λ = ( l 1 , . . . , l n ) ∈ Λ , t �→ t λ := t l 1 1 · · · t l n t = ( t 1 , . . . , t n ) ∈ � T ; n , Λ( V ) = { eigenweights of � T � V } ; P = conv Λ( V ) the weight polytope; ∆ = ∆ G ⊂ Λ the set of roots (= nonzero T -eigenweights of Lie G ), ∆ = ∆ + ⊔ ∆ − (positive and negative roots); W = N G ( T ) /T the Weyl group; ( · , · ) a W -invariant inner product on Λ; C = C G = { λ ∈ Λ ⊗ Z Q | ( λ, α ) ≥ 0 , ∀ α ∈ ∆ + } the positive Weyl chamber. It is a fundamental domain for W � Λ ⊗ Z Q . 4

2. Orbits For each face F of P (of any dimension, including P itself) let: V F ⊆ V be the span of � T -eigenvectors with weights in F ; V ′ T -stable complement of V F ; V = V F ⊕ V ′ F ⊆ V be the � F ; E F = projector V → V F . Theorem 1. There is a 1-1 correspondence: ( G × G ) -orbits Y ⊆ X ← → faces F ⊆ P , (int F ) ∩ C � = ∅ . Orbit representatives are: Y ∋ y = � E F � . Stabilizers are computed. � � ∆ \ �F� ⊥ � � � Dimensions: dim Y = dim F + � . Partial order: Y 1 ⊂ Y 2 ⇐ ⇒ F 1 ⊂ F 2 . 5

X = P (Mat n ) = P n 2 − 1 . Example 2. G = PGL n , V = C n ⇒ = Here � G = SL n , T = { diagonal matrices } , Λ( V ) = { ε 1 , . . . , ε n } = the standard basis of Z n , ∆ + = { ε i − ε j | i < j } , ∆ = { ε i − ε j | i � = j } , C = { λ = ( l 1 , . . . , l n ) | l 1 ≥ · · · ≥ l n } We see that P = conv { ε 1 , . . . , ε n } is a simplex, the faces of P whose interior intersects C are F r = conv { ε 1 , . . . , ε r } , r = 1 , . . . , n, the respective projectors are E F r = diag(1 , . . . , 1 , 0 , . . . , 0) , � �� � r and the orbits are Y r = P (matrices of rank r ) . 6

Proof. Step 1. Let K ⊂ G be a maximal compact subgroup; then G = KTK (Cartan decomposition) ⇒ X = KTK ⇒ = = T intersects all ( G × G )-orbits in X . Step 2. By toric geometry, there is a 1-1 correspondence: T -orbits in T ← → all faces F ⊆ P , which respects partial order, � E F � being the orbit representatives. Step 3. Compute the stabilizers of � E F � in G × G . In particular, this yields the orbit dimensions. Step 4. Given y = � E F � ∈ Y = GyG , the structure of ( G × G ) y implies that Y diag T = � w 1 ,w 2 ∈ W Tw 1 yw 2 ? It follows that Y 1 = Y 2 ⇐ ⇒ F 1 = w F 2 for some w ∈ W. There exists a unique F + = w F such that (int F + ) ∩ C � = ∅ . 7

3. Local structure Fix a closed orbit Y 0 ⊂ X . What is the structure of X in a neighborhood of Y 0 ? W.l.o.g. V is assumed to be a multiplicity-free G -module. By Theorem 1, Y 0 = Gy 0 G , y 0 = � E λ 0 � , λ 0 ∈ P a vertex, E λ 0 is the projector V = C v λ 0 ⊕ V ′ λ 0 → C v λ 0 , where v λ 0 is the (unique) eigenvector of weight λ 0 (a highest weight vector). Associated parabolic subgroups: P + = G � v λ 0 � , P − ⊆ G . Levi decomposition: P ± = P ± u ⋊ L , L = P + ∩ P − ⊇ T . λ 0 } is a ( P − × P + ) - ∈ V ′ Theorem 2. ˚ X = { x = � A � ∈ X | Av λ 0 / stable neighborhood of y 0 in X . X ≃ P − u × Z × P + where Z = L ⊆ End( V ′ ˚ u , λ 0 ⊗ ( − λ 0 )) . 8

Proof is based on the local structure of P − � V in a neighborhood of v λ 0 (Brion, Luna, Vust). Remark. Z is a reductive algebraic semigroup with 0 (corre- sponding to y 0 ), called the slice semigroup. Regular case: λ 0 ∈ int C = ⇒ L = T ⇒ Z affine toric variety. = Example 3. In the notation of Example 2, Y 0 = P (matrices of rank 1) , λ 0 = ε 1 , v λ 0 = e 1 , 1 ∗ 1 0 · · · 0 ∗ 0 · · · 0 . ∗ P + 0 . E ∗ E 0 P − u = , u = , L = ≃ GL n − 1 , . . . . 0 0 V ′ λ 0 = � e 2 , . . . , e n � , Z = Mat n − 1 . 9

4. Normality Does X have normal singularities? It suffices to consider singu- larities in a neighborhood of a closed orbit Y 0 . By Theorem 2 it suffices to study the singularity of Z at 0. L is reductive = ⇒ L -modules are completely reducible . Simple L -modules V = V L ( λ ) ← → highest weights λ ∈ Λ ∩ C L , ∈ Λ( V ) , ∀ α ∈ ∆ + λ + α / L . V L ( λ ) ⊗ V L ( µ ) ≃ V L ( λ + µ ) ⊕ · · · ⊕ V L ( λ + µ − α 1 − · · · − α k ) ⊕ · · · ( α i ∈ ∆ + L ) . Definition. Weights µ 1 , . . . , µ k L -generate a semigroup Σ ⊂ Λ if Σ = { µ | V L ( µ ) ֒ → V L ( µ i 1 ) ⊗ · · · ⊗ V L ( µ i N ) } . Observation: “generate” (in a usual sense) = ⇒ “ L -generate”; ⇐ = fails in general. 10

Theorem 3. Let λ 0 , λ 1 . . . , λ m be the highest weights of the simple G -submodules in V and α 1 , . . . , α r be the simple roots in ∆ + G \ ∆ + L . Put Σ = (director cone of P ∩ C at λ 0 ) ∩ Λ . Consider the following conditions: (1) X is normal along Y 0 ; (2) T is normal at y 0 ; (3) Σ is L -generated by λ i − λ 0 , − α j . (4) Σ is generated by λ − λ 0 , ∀ λ ∈ Λ( V ) . Then (1) ⇐ ⇒ (3) = ⇒ (2) ⇐ ⇒ (4) . ⇐ ⇒ λ i − λ 0 , − α j L -generate a saturated semi- Remark. (3) group. 11

0 ( � 2 Example 4. G = Sp 4 , V = S 3 C 4 ⊕ S 2 0 C 4 ), the highest weights { λ 0 = 3 ω 1 , λ 1 = 2 ω 2 } ( ω i denote the fundamental weights, α i the simple roots). Here L ≃ SL 2 × C × , ∆ L = {± α 2 } . λ 1 − λ 0 , − α 1 L -generate { bold dots } (Clebsch–Gordan formula). ⇒ X non-normal along Y 0 ; becomes normal if we add λ 2 = 2 ω 1 . = ✟ ❍❍❍❍❍❍❍❍ s ✟ � ✟ ✟ � ✟ s s s s ✇ ✇ ✇ ✇ α 2 ✟ � ✟ ✟ � ✟ ❍ λ 1 s s s s ✇ � ✁ ✻ ❆ ✁ ❆ � P ✁ ❆ C s s s s ✇ ✇ ✇ ✇ ω 2 � ✁ ❆ ❍ ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ � ❍ ✁ ❆ ❍ s s s � s s ❍ ✁ ❆ ❍ � ❍ ✁ ❆ ❍ ❦ � s s s s ✇ ✇ ❍ ω 1 λ 0 ✁ ❆ ❍ � ❍ ✁ � ❆ ❍ ❍ s s s s s s ✇ ❆ ❅ ✁ ❍ ❍ ❆ ❅ ✁ ❍ ✲ ✇ ❆ ❅ ✁ α 2 ❆ ❅ ✁ ❅ ❘ s s s s s α 1 ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❍ ✟ s ❍ s s ✟✟✟✟✟✟✟✟ s s ❍ ❍ ❍ ❍ ❍ ❍ ❍ s 12

5. Smoothness Theorem 4. X is smooth along Y 0 ⇐ ⇒ (1)&(2)&(3) : (1) L = GL n 1 × · · · × GL n p . (2) L � V ⊗ ( − λ 0 ) is polynomial. ֓ [ GL n i � C n i ] , ∀ i . (3) [ L � V ⊗ ( − λ 0 )] ← Remark. (1), (2), (3) are reformulated in terms of Λ( V ). Idea of the proof. X is smooth ⇐ ⇒ Z is smooth ⇐ ⇒ Z ≃ Mat n 1 × · · · × Mat n p . Example 5. G = SO 2 m +1 , V = V G ( ω i ) ( ω i are the fundamental weights). a) i < m : V = � i C 2 m +1 , L ≃ GL i × SO 2 m +1 − 2 i ⇒ X singular. = b) i = m : V = spinor module ≃ � • C m ⊗ ω m over L ≃ GL m ⇒ = X smooth. 13

6. “Small” compactifications Let G be a simple Lie group. We take a closer look at X = G ⊆ P (End V ) for “small” V = V G ( ω i ) ( ω i are the fundamental weights) or V = Lie G (adjoint representation). Results: 1) ( G × G )-orbits, their dimensions, Hasse diagrams of partial order. 2) Non-normal: ( SO 2 m +1 , ω i ), i < m ; ( Sp 2 m , ω m ); ( G 2 , ω 2 ); ( F 4 , ω i ), i = 3 , 4. 3) Smooth: ( SL n , ω i ), i = 1 , n − 1; ( SL n , Ad), n ≤ 3; ( SO 2 m +1 , ω m ); ( Sp 4 , ω 1 ); ( Sp 4 , Ad); ( G 2 , ω 1 ). 14