Reductive group actions and some problems concerning their quotients

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1 Reductive group actions and some problems concerning their quotients Brandeis University January 2014

2 Linear Algebraic Groups A complex linear algebraic group G is an affine variety such that the mappings G G G, (g, h) gh and G G, g g 1 are morphisms. Equivalently, G is isomorphic to a closed subgroup (Zariski topology) of some GL n (C). Examples are GL n (C), SL n (C), SO n (C). Any finite group. Any torus (C ) m. Can think G finite for whole lecture if you want. A representation of G is a homomorphism τ : G GL(V ) for some (finite dimensional) complex vector space V. Call V a G-module. So τ is a group homomorphism and a morphism of affine varieties. V is irreducible if the only G-stable subspaces are {0} and V. If V is a direct sum of irreducible G-submodules, say V is completely reducible. G is reductive if every G-module is completely reducible. Then GL n (C), SO n (C), G finite, G a torus are all reductive.

3 Orbit spaces Often some important object in mathematics is parameterized by the elements of V and the objects associated to v and gv are isomorphic. So one wants to know the isomorphism classes, i.e., the orbits Gv = {gv g G}. Consider irreducible hypersurfaces of degree d in P n (C). These are the zeroes H f of some irreducible f homogeneous of degree d in R := C[x 0, x 1,..., x n ]. The group G = GL n+1 (C) acts on R d sending f to f g 1, g G. Also, G induces all automorphisms of P n (C). Then H f and H f are isomorphic under an automorphism of P n (C) if and only if f and f are in the same orbit of G. One approach to finding the orbits is to compute the invariants of the action of G. Let V be a G-module where G is reductive. C[V ] G = {f C[V ] f (gv) = f (v) for all g, v}.

4 Hilbert s Theorem Theorem(Hilbert): C[V ] G is finitely generated. Let p 1,..., p d C[V ] G be generators, let I O(C d ) be their ideal of relations. So h I iff h(p 1,..., p d ) 0. p := (p 1,..., p d ): V Z := Var(I). Then p is onto. Each fiber of p contains a unique closed G-orbit. So Z {closed G-orbits in V }. Ex: G = C, V = C 2, t (x, y) = (tx, t 1 y), t C, (x, y) V. Then p = xy : V C = Z. If z 0, then p 1 (z) = {(tz, t 1 )} is a (closed) orbit. Suppose that z = 0, i.e., xy = 0. Then we have three orbits: {(x, 0) x 0}, {(0, y) y 0} and {(0, 0)}. Cannot separate these orbits using the invariants. But one of the orbits is closed.

5 The stratification and our questions Let Gv be closed. Then H := G v is reductive. Let (H) be conjugacy class H and Z (H) = {closed orbits with isotropy group in (H)} (considered as subset of Z). The Z (H) are a finite stratification of Z by locally closed smooth irreducible subvarieties. The spaces Z and their stratifications are quite complicated. Some questions about them. Let ϕ Aut(Z). Does ϕ preserve the stratification? Question of Kuttler+Reichstein. Does ϕ preserve each stratum? Is there a lift Φ: V V of ϕ? Means p(φ(v)) = ϕ(p(v)), v V. Can arrange Φ-equivariant? Means Φ(gv) = gφ(v) for all g, v. Ex: C on C 2. Z = C has Z ({e}) = C \ {0} and Z (C ) = {0}. Looking at Z = C we don t see the strata. Everything false.

6 Tori V = 2C 2, C -action t(x, y, x, y ) = (tx, t 1 y, tx, t 1 y ). p = (p 1,..., p 4 ) = (xy, xy, x y, x y ): V C 4. Z = Var(y 1 y 4 y 2 y 3 ). Strata are Z \ {0} and {0} where {0} is the singular point of Z. Any ϕ Aut(Z) preserves the strata. First three questions have a positive answer. But lift Φ may not be equivariant. Let σ(t) = t 1, t C, an automorphism of C. Let Φ: 2C 2 2C 2 interchanging x and y, x and y. Then Φ(tv) = σ(t)φ(v). We say that Φ is σ-equivariant. Best one can hope for in a lift. In general, let Z pr denote the principal (the open) stratum of Z and V pr = p 1 (Z pr ). We say that V is k-principal if codim V \ V pr is k and V pr consists of closed G-orbits. Our representation above is 2-principal.

7 Tori Theorem: Assume that V is 2-principal and that G 0 is a torus. Let ϕ Aut(Z). Then there is a σ-equivariant lift Φ of ϕ where σ is an automorphism of G. Since Φ is σ-equivariant, ϕ sends Z (H) to Z (σ(h)). Preserves stratification. Suppose (V 1, G 1 ) and (V 2, G 2 ) as in Theorem. If Z 1 Z 2, then V 1 V 2 sending G 1 to G 2. Proof: (V, G) = (V 1 V 2, G 1 G 2 ), Z Z 1 Z 2, ϕ: Z Z interchanges Z 1 and Z 2. Then Φ interchanges V 1 and V 2 and Φ (0) gives the isomorphisms. Similar results for ϕ Aut H (Z).

8 Preserving the stratification What of general G? Does Aut(Z) preserve the stratification (is stratif. intrinsic)? Question asked by Kuttler and Reichstein. Theorem: Suppose that V is 3-principal or 2-principal and orthogonal (G O(V )). Then the stratification is intrinsic. Theorem: Suppose that G is simple and V G = (0). Then there are at most finitely many V, up to isomorphism, which are not 3-principal. Hence stratification of Z almost always intrinsic. Similar statement for G semisimple.

9 Lifting Automorphisms C acts on Z by t p(v) = p(tv). p = (p 1,..., p d ). We may assume that p i is homogeneous, of degree e i, and then t (y 1,..., y d ) = (t e 1 y 1,..., t e d y d ) for (y 1,..., y d ) Z. Assume V G = (0), harmless. So {p(0)} is the stratum Z (G). Let ϕ Aut(Z) which preserves the stratification. Say that ϕ is deformable if ϕ 0 := lim t 0 t 1 ϕ(t z) exists for all z Z. If e 1 = = e d, then ϕ 0 is an ordinary derivative. ϕ 0 is quasilinear, i.e., ϕ 0 t = t ϕ 0, t C. If ϕ 1 also deformable, then ϕ 0 is invertible. Write ϕ 0 Aut ql (Z). A linear algebraic group. We say that V is admissible if every ϕ is deformable. Theorem: If G simple, all but finitely many V admissible (V G = (0)), up to isomorphism. If V 2-principal, G 0 torus, V is admissible.

10 The Lifting Property We say that V has the lifting property if every ϕ Aut ql (Z) has a lift to GL(V ). Suppose that V is 2-principal and stratification of Z is intrinsic. Then V has the lifting property iff N GL(V ) (G) maps onto Aut ql (Z). Elements of N GL(V ) (G) are σ-equivariant. If V is admissible, GL(V ) G maps onto Aut ql (Z) 0, so problem is component group of Aut ql (Z). Theorem: Let ϕ Aut H (Z) where V is admissible. If V has the lifting property, then there is a holomorphic σ-equivariant lift of ϕ.

11 Representations containing the adjoint representation Theorem (Kuttler). Let V = rg, direct sum of r copies of g. If V is 4-principal, then every ϕ Aut(Z) has an algebraic lift. The lift is not necessarily an automorphism nor σ-equivariant. Theorem. V is 2-principal is enough. Best possible. Moreover, V is admissible, so that any ϕ Aut H (Z) has a σ-equivariant holomorphic lift. Hence ϕ permutes the strata sending Z (H) to Z (σ(h)). Extension of first theorem. Write G 0 = G s S where G s is semisimple and S is a (central) torus. Theorem. Suppose that V is 4-principal and (V, G s ) contains g s. Then any ϕ Aut(Z) lifts to some algebraic Φ.

12 Classical Groups Theorem: Let (V, G) be one of the following representations. Then V is admissible and V has the lifting property. 1. (kc n l(c n ), SL n ), n 3 where k + l 2n and if kl = 0, then k + l > 2n. 2. (kc n l(c n ), GL n ) where k, l > n. 3. (kc n, SO n ) where k n (kc n, O n ) where k > n (kc 2n, Sp 2n ) where k 2n + 2, n (kc 7, G 2 ) where k > (kc 8, Spin 7 ) where k > 5. Admissible and don t have lifting property: 1. (2nC n, SL n ), n (4C 7, G 2 ). 3. (5C 8, Spin 7 ).