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Representation Theory and Cohomology Intertwining Representa- tion Theory and Cohomology Eric M. Friedlander Representation Theory and Cohomology Support varieties for algebraic groups Eric M. Friedlander July 7, 2016 Emil Artin Lecture, Heidelberg Emil Artin Intertwining Representa- tion Theory and Cohomology Eric M. Friedlander Support varieties for algebraic groups Michael Artin Intertwining Representa- tion Theory and Cohomology Eric M. Friedlander Support varieties for algebraic groups Representation theory Intertwining Representa- tion Theory and Cohomology G a group and V a vector space Eric M. Friedlander Support varieties for G × V ! V ; (g; v) 7! g ◦ v algebraic groups Conditions: (g1 · g2) ◦ v = g1 ◦ (g2 ◦ v); g ◦ (a · v + b · w) = a · (g ◦ v) + b · (g ◦ w). What sort of groups? What sorts of vector spaces? Lie theory Intertwining Representa- tion Theory Sophus Lie and Cohomology 1842 - 1899 Eric M. Friedlander Perspective of geometry and Support varieties for differential equations algebraic groups \continuous transformation groups" acting continuously, e.g. on a real or complex vector space V Lie theory: Understand these continuous representations of G in terms of representations of g = Lie(G). Algebraic groups Intertwining Representa- tion Theory and Claude Chevalley Cohomology Eric M. Friedlander 1923 - 2003. Support varieties for algebraic Algebraic groups groups over a field k GLn is zero locus n2+1 inside A of det(xi;j ) · z = 1 Fact: Each simple complex Lie group can be viewed as zero locus of further polynomial equations inside some GLN (C). Algebraic representations Intertwining Representa- tion Theory What is an algebraic representation of an algebraic group? and Cohomology ⊕N Eric M. If V is finite dimensional, V = k , then G × V ! V is Friedlander algebraic (a.k.a. \rational representation") if each matrix Support coefficient as a function of G is algebraic (i.e., in k[G]). varieties for algebraic groups Example: Let G = GL2 act on polynomials of degree n in 2 variables k[x; y]n. Explicitly, we can write this as a b ◦ xi y n−i = (ax + by)i (cx + dy)n−i c d Equivalently: comodule structure∆: V ! V ⊗ k[G], so that P P ∆(v) = vi ⊗ fi with g ◦ v = i fi (g)vi . characteristic p > 0 Intertwining Representa- tion Theory and Example: Fp { field of p-elements. (Every non-zero element Cohomology has an inverse; add 1 to itself p times and the answer is 0.) Eric M. d Friedlander Example: q = p ; d > 0. There is a unique finite field Fq of order q. Support varieties for Example: If F is a field of characteristic p > 0, then so is F (x). algebraic groups N If X ⊂ A is the zero locus of polynomial equations with N coefficients in Fq, then sending (x1;:::; xN ) 2 A to q q N (x1 ;:::; xN ) 2 A sends points of X to points of X . Key point:( a + b)p = ap + bp in characteristic p. Frobenius map F q : X ! X . Wildness for finite groups, char p > 0 Intertwining Representa- k-linear actions of =p × =p on k-vector space V correspond tion Theory Z Z and to actions of Cohomology Eric M. p p p p Friedlander k[g; h]=(g = 1 = h ) ' k[x; y]=(x ; y ); g = x+1; h = y+1 Support varieties for Example: Indecomposable, not irreducible algebraic groups v v v vn−1 v •1 •2 •3 ::: • •n y x y x y x •Õ •Õ ::: •Õ y x y x ••Ô ::: • Ò Z=p × Z=p has wild representation type (for p > 2). Solomon Lefschetz Intertwining Representa- tion Theory and Cohomology Solomon Lefschetz Eric M. Friedlander Support 1884 - 1972 varieties for algebraic groups Applications of algebraic topology to algebraic geometry (classical) algebraic geometry Characteristic p STINKS! Cohomology Intertwining Representa- tion Theory and Definition Cohomology Eric M. If G × M ! M is a G-action on the k-vector space M, then Friedlander 0 G Support H (G; M) = M = fm 2 M; g ◦ m = m; 8g 2 Gg: varieties for algebraic i i 0 groups H (G; M) = (R (H (G; −))(M): If every indecomposable G-module is irreducible, then Hi (G; M) = 0; i > 0. H1(G; M) equals the group of equivalence classes of short exact sequences 0 ! M ! E ! k ! 0 of G-modules (i.e., extensions of k by M). Cohomology and Geometry Intertwining Representa- tion Theory and Daniel Quillen Cohomology Eric M. Friedlander 1940 { 2011 Support varieties for algebraic Spectrum of cohomology of a groups finite group G Spec(H∗(G; k)), affine algebraic variety Extension of Quillen by J. Carlson et al: use spirit of Quillen to study representations of a finite group G. Failure of Lie theory over fields of characteristic p > 0 Intertwining Representa- tion Theory and Cohomology EXAMPLE SL2 action on the homogeneous polynomials of Eric M. degree precisely p in two variables: V = k[x; y]p. Inside V , Friedlander there is a 2-dimensional subrepresentation W ⊂ V consisting of Support polynomials linear in xp; y p. There is no splitting of W ⊂ V as varieties for algebraic representations of SL2. groups V is indecomposable, but not irreducible. Much WORSE news: If action of SL2 on vector space V factors through F : SL2 ! SL2, then Lie algebra action is trivial (because differential d(F ) = 0 : sl2 ! sl2). Functors and group schemes Intertwining Representa- tion Theory and Alexander Grothendieck Cohomology Eric M. Friedlander 1928 - 2014 Support varieties for Functorial point of view algebraic groups A group scheme over k is a functor (comm. k-alg) to (groups). Replace the Lie algebra of G by “infinitesimal neighborhoods of the identity", so called \Frobenius kernels" G(r). Frobenius kernels Intertwining Representa- tion Theory One can view G(r) ⊂ G as a representable subfunctor of G and Cohomology r Eric M. (comm: k − alg) ! (groups); R 7! kerfF : G(R) ! G(R)g: Friedlander Support varieties for algebraic Example groups pr GLN(r) has coordinate algebra k[Xi;j ]=(Xi;j − δi;j ), a finite dimensional, commutative, local k-algebra; mulitplication of dual kGLN(r) is given by the multiplication of GLN . kG(r) is always a f. dim, co-commutative Hopf algebra. Given a representation G × V ! V , this structure is faithfully reflected by the collection of structures fG(r) × V ! V g. The additive group Ga Intertwining Representa- tion Theory and Definition Cohomology + a: (comm. k-alg) ! (abelian groups); a(R) = R . Eric M. G G Friedlander k[Ga] = k[T ]; group structure determined by comultiplication Support varieties for ∆ : k[T ] ! k[T ] ⊗ k[T ]; T 7! (T ⊗ 1) + (1 ⊗ T ): algebraic groups Lemma A Ga-action on a k-vector space V is naturally equivalent to the following data: An infinite sequence of p-nilpotent, pairwise commuting: operators u0; u1; u2; u3 ::: : V ! V such that for any v 2 V all but finitely many ui (v) are 0. Varieties for Ga-modules Intertwining Representa- tion Theory and Definition Cohomology The cohomological variety V coh( ) ≡ Spec H∗( ; k). Eric M. Ga cont Ga Friedlander The 1-parameter subgroup variety V (Ga) ≡ f : Ga ! Gag. Support varieties for algebraic groups Proposition coh 1 V (Ga) ' A ' V (Ga): Definition coh ∗ ∗ V (Ga)M ≡ fp ⊂ H (Ga; k): p ⊃ ann(H (Ga; M)g. V (Ga)M ≡ f : Ga ! Ga : such that NOT all blocks of size p for action at g. Support varieties for Ga Intertwining Representa- tion Theory and coh 1 Cohomology For M finite dimensional, V (Ga)M = V (Ga)M ⊂ A . Eric M. Friedlander Many \standard" properties including V (Ga)M = f0g if M 1 Support is injective, V (Ga)M = A if M = k. varieties for algebraic \Mock injective" modules: there exist (necessarily infinite groups dimensional) Ga-modules M which are not injective, but V (Ga)M = f0g. 1 Know exactly which subvarieties X ⊂ A are of form X = V (Ga)M for some finite dimensional Ga-module M. 1 Lots of interesting questions about which X ⊂ A are of form X = V (Ga)M for an arbitrary Ga-module M. Other algebraic groups G Intertwining Representa- tion Theory and Cohomology Eric M. Friedlander Cohomology NOT very useful in general. For example, H∗(G; k) is trivial for G a simple algebraic group. Support varieties for algebraic groups Will describe a theory using 1-parameter subgroups : Ga ! G which has many useful properties. For G unipotent (e.g., UN ⊂ GLN ), then study of 1-parameter subgroups leads to cohomological calculations. 1-parameter subgroups for infinitesimal kernels G(r) Intertwining Representa- tion Theory Andrei Suslin and Cohomology Eric M. Joint work: Quillen's geometry Friedlander extends to Frobenius kernels. Support varieties for ∗ algebraic computations for H (G(r); k) groups in terms of the variety of V (G(r)) of infinitesimal 1-parameter subgroups Ga(r) ! G(r). coh Theorem [Suslin-F-Bendel] V (G(r))M can be identified with the variety V (G(r))M . Action of G on M at a 1-parameter subgroup Intertwining Representa- Theorem tion Theory and Assume that G is a linear algebra group of exponential type. Cohomology The ind-variety V (G) of 1-parameter subgroups of G is ' Eric M. Friedlander variety C1(Np(Lie(G))) consisting of finite sequences of Support p-nilpotent, pair-wise commuting elements of Lie(G): varieties for algebraic ∼ groups fB 2 C1(Np(Lie(G)))g ! V (G); B 7! EB : Definition The action on a rational G-module M at the 1-parameter subgroup EB : Ga ! G is the action of the p-nilpotent operator X (EBs )∗(us ) 2 kG: s≥0 Formulation of support variety V (G)M Intertwining Representa- Definition tion Theory and The support variety V (G)M ⊂ V (G) of M is the subset of Cohomology those B 2 C1(Np(Lie(G))) such that B has some block of Eric M.
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