David Vogan. 40th Anniversary Midwest Representation Theory Conference. In honor of the 65th birthday of Rebecca Herb and in memory of Paul Sally, Jr.

Size: px

Start display at page:

Download "David Vogan. 40th Anniversary Midwest Representation Theory Conference. In honor of the 65th birthday of Rebecca Herb and in memory of Paul Sally, Jr."

Robert Hall
5 years ago
Views:

1 Department of Mathematics Massachusetts Institute of Technology Local Local : reprise 40th Anniversary Midwest Representation Theory Conference In honor of the 65th birthday of Rebecca Herb and in memory of Paul Sally, Jr. The University of Chicago, September 5 7, 2014

2 Outline Local Local Local : reprise Local : reprise

3 Adeles Arithmetic problems matrices over Q. { ( ) } 1 0 Example: count v Z 2 t v v N. 0 1 Hard: no analysis, geometry, topology to help. Possible solution: use Q R. Example: find area of { v R 2 t v Same idea with Q Q p leads to ( ) } 1 0 v N. 0 1 Local Local : reprise A = A Q = R Q p = Q v, p v {p, } locally compact ring Q discrete subring. Arithmetic analysis on GL(n, A)/GL(n, Q).

4 Background about GL(n, A)/GL(n, Q) Gelfand: analysis re G irr (unitary) reps of G. analysis on GL(n, A)/GL(n, Q) irr reps π of π = v {p, } v {p, } GL(n, Q v ) π(v), π(v) GL(n, Q v ) Local Local : reprise Building block for harmonic analysis is one irr rep π(v) of GL(n, Q v ) for each v. Contributes to GL(n, A)/GL(n, Q) tensor prod has GL(n, Q)-fixed vec.

5 Local Big idea from unpublished paper: GL(n,? Q v ) n-diml reps of Gal(Q v /Q v ). Big idea actually goes back at least to 1967; 1973 paper proves it for v =. (LLC) Caveat: need to replace Gal by Weil-Deligne group. Caveat: Galois reps in (LLC) not irr. Caveat: Proof of (LLC) for finite v took another 25 years (finished 2 by Harris 3 and Taylor 2001). Conclusion: irr rep π of GL(n, A) one n-diml rep σ(v) of Gal(Q v /Q v ) for each v. Local Local : reprise 1 Paul Sally did not believe that big idea and unpublished belonged together. In 1988 he arranged publication of this paper. 2 History: finished by HT is too short. But Phil s not here, so... 3 Not that one, the other one.

6 Background about arithmetic {Q 2, Q 3,..., Q } loc cpt fields where Q dense. If E/Q algebraic extension field, then E v = def E Q Q v is a commutative algebra over Q v. E v is direct sum of algebraic extensions of Q v. If E/Q Galois, summands are Galois exts of Q v. Γ = Gal(E/Q) transitive on summands. Choose one summand E ν E Q Q v, define Local Local : reprise Γ v = Stab Γ (E ν ) = Gal(E ν /Q v ) Γ. Γ v Γ closed, unique up to conjugacy. Conclusion: n-diml σ of Γ n-diml σ(v) of Γ v. Čebotarëv: knowing almost all σ(v) σ.

7 Global conjecture Write Γ = Gal(Q/Q) Gal(Q v /Q v ) = Γ v. analysis on GL(n, A)/GL(n, Q) irr reps π of v {p, } GL(n, Q v ) π GL(n,Q) 0 π = π(v), π GL(n,Q) 0 v {p, } Local Local : reprise LLC n-diml rep σ(v) of Γ v, each v which σ(v)?? GLC: π GL(n,Q) 0 if reps σ(v) of Γ v one n-diml representation σ of Γ. If Γ finite, most Γ v = g v cyclic, all g v occur. Arithmetic prob: how does conj class g v vary with v?

8 Starting local for GL(n, R) All that was why it s interesting to understand GL(n, R) LLC n-diml reps of Gal(C/R) n-diml reps of Z/2Z { } n n cplx y, y 2 = Id /GL(n, C) conj : more reps of GL(n, R) (Galois Weil). But what have we got so far? y m, 0 m n (dim( 1 eigenspace)) m unitary char ξ m : B {±1}, ξ m (b) = sgn(b jj ) unitary rep π(y) = Ind GL(n,R) B ξ m. j=1 Local Local : reprise This is all irr reps of infl char zero.

9 Integral infinitesimal characters Infinitesimal char for GL(n, R) is unordered tuple (γ 1,..., γ n ), (γ i C). Assume first γ integral: all γ i Z. Rewrite γ = ( ) γ 1,..., γ }{{ 1,..., γ } r,..., γ r (γ }{{} 1 > > γ r ). m 1 terms m r terms Local Local : reprise A flat of type γ consists of 1. flag V = {V 0 V 1 V r = C n }, dim V i /V i 1 = m i ; 2. and the set of linear maps F = {T End(V ) TV i V i, T Vi /V i 1 = γ i Id}. Such T are diagonalizable, eigenvalues γ. Each of V and F determines the other (given γ). param of infl char γ = pair (y, F) with F a flat of type γ, y n n matrix with y 2 = Id.

10 Integral local for GL(n, R) γ = ( ) γ 1,..., γ }{{ 1,..., γ } r,..., γ }{{} r m 1 terms m r terms (γ 1 > > γ r ) ints. parameter of infl char γ = pair (y, V), y 2 = Id, V flag, dim V i /V i 1 = m i. π GL(n, R), infl char γ LLC {(y, V}/conj by GL(n, C). Local Local : reprise So what are these GL(n, C) orbits? Proposition Suppose y 2 = Id n and V is a flag in C n. There are subspaces P i, Q i, and C ij (i j) s.t. 1. y Pi = +Id, y Qi = Id. 2. y : C ij Cji. 3. V i = i i (P i + Q i ) + i i,j C i,j. 4. p i = dim P i, q i = dim Q i, c ij = dim C ij = dim C ji depend only on GL(n, C) (y, V).

11 Action of involution y on a flag Last i rows represent subspace V i in flag. Arrows show action of y. V 3 V 2 P 3 +1 Q 3 1 C 31 P 2 +1 Q 2 1 C 21 V 1 P 1 +1 Q 1 1 C 12 C 13 Represent diagram symbolically (Barbasch) ( γ + 1,..., γ+ 1, γ 1,..., γ 1,..., γ r +,... }{{} dim P 1 terms }{{} dim Q 1 terms }{{} dim P r terms ) (γ 1 γ 2 ),..., (γ 1 γ 2 ),..., (γ r 1 γ r ),... }{{}}{{} dim C 12 terms dim C r 1,r terms This is involution in S n plus some signs., γr,... }{{}, dim Q r terms Local Local : reprise

12 General infinitesimal characters Recall infl char for GL(n, R) is unordered tuple (γ 1,..., γ n ), (γ i C). Organize into congruence classes mod Z: γ = ( γ 1,..., γ n1, γ n1 +1,..., γ n1 +n }{{} 2,..., }{{} cong mod Z cong mod Z ) γ n1 + +n s 1 +1,..., γ n, }{{} cong mod Z then in decreasing order in each congruence class: γ = ( γ1, 1..., γ1 1 1,..., γr }{{} 1,..., γr 1 }{{} 1,..., γ1, s..., γ1 s s,..., γr }{{} s,..., γr s ) }{{} s m 1 1 terms m 1 r 1 terms } {{ } n 1 terms m s 1 terms m 1 rs terms } {{ } n s terms γ 1 1 > γ 1 2 > > γ 1 r 1, γ s 1 > γ s 2 > > γ s r s. Local Local : reprise

13 Nonintegral flats Start with general infinitesimal character γ = ( γ1, 1..., γ1 1 1,..., γr }{{} 1,..., γr 1 }{{} 1 m 1 1 terms m 1 r 1 terms } {{ } n 1 terms A flat of type γ consists of,..., γ1, s..., γ1 s }{{} m s 1 terms s,..., γ r s,..., γr s }{{} s m 1 rs terms } {{ } n s terms 1a. direct sum decomp C n = V 1 V s, dim V k = n k ; 1b. flags V k = {V0 k Vr k k = V k }, dim Vi k /Vi 1 k = mi k ; 2. and the set of linear maps F({V k }, γ) = {T End(C n ) TVi k Vi k, T V k i /V i 1 k = γi k Id}. Such T are diagonalizable, eigenvalues γ. Each of (1) and (2) determines the other (given γ). invertible operator e(t ) = def exp(2πit ) depends only on flat: eigenvalues are e(γ k i ) (ind of i), eigenspaces {V k }. param of infl char γ = pair (y, F) with F a flat of type γ, y n n matrix with y 2 = e(t ). ) Local Local : reprise

14 parameters for GL(n, R) Infl char γ = (γ 1,..., γ n ) (γ i C unordered). Recall parameter (y, F) is 1. direct sum decomp of C n, indexed by {γ i + Z}; 2. flag in each summand 3. y GL(n, C), y 2 = e(γ i ) on summand for γ i + Z. Proposition GL(n, C) orbits of parameters of infl char γ are indexed by 1. pairing some (γ i, γ j ) with γ i γ j Z 0; and 2. labeling each unpaired γ k with + or. Example infl char (3/2, 1/2, 1/2): [(3/2, 1/2), ( 1/2) ± ], two params [(3/2, 1/2), (1/2) ± ], two params [(1/2, 1/2), (3/2) ±, two params [(3/2) ±, (1/2) ±, ( 1/2) ± ] eight params [(γ 1, γ 2 )] disc ser, HC param γ 1 γ 2 of GL(2, R) [γ + or ] character t t γ (sgn t) 0 or 1 of GL(1, R). Local Local : reprise

15 Other reductive groups G(R) real red alg group, G dual (cplx conn red alg). Semisimple conj class H g infl char. for G. For semisimple T g and integer k, define g(k, T ) = {X g [T, X] = kx}. Say T T if T T + k>0 g(k, T ). Flats in g are the equivalence classes (partition each semisimple conjugacy class in g). Exponential e(t ) = exp(2πit ) G const on flats. If G(R) split, parameter for G(R) is (y, F) with F g flat, y G, y 2 = e(f). Theorem (LLC, 1973) Partition Ĝ(R) into finite L-packets G orbits of (y, F). Infl char of L-packet is G F. Future ref: (y, F) inv w(y, F) W. For infl char 0, Theorem says irr reps partitioned by conj classes of homs Gal(C/R) G. Local Local : reprise

16 and now for something completely different... G cplx conn red alg group. Problem: of G/(equiv)? Soln (): {x Aut(G x 2 = 1}/conj. Details: given aut x, choose cpt form σ 0 of G s.t. xσ 0 = σ 0 x = def σ. Example. ( ) Ip 0 G = GL(n, C), x p,q (g) = conj by. 0 I q Choose σ 0 (g) = t g 1 (real form U(n)). ( ) ( A B t ) A σ p,q = t 1 C C D t B t, D real form U(p, q). Another case of matrices almost of. Local Local : reprise

17 involutions G cplx conn red alg group. Galois parameter is x G s.t. x 2 Z (G). θ x = Ad(x) Aut(G) involution. Say x has central cochar z = x 2. ( ) e( q/2n)ip 0 G = SL(n, C), x p,q =. 0 e(p/2n)i q x p,q real form SU(p, q), central cocharacter e(p/n)i n. Theorem () Surjection {Galois params} {equal rk of G(C)}. ( ) Ip 0 G = SO(n, C), x p,q = allowed iff q = 2m even. 0 I q x n 2m,m real form SO(n 2m, 2m), central cochar I n. ( ) 0 1 G = SO(2n, C), J =, x 1 0 J = n copies of J on diagonal. x J real form SO (2n), central cochar I 2n. Local Local : reprise

18 Imitating Since Galois param part of param, why not complete to a whole param? Start with z Z (G) Choose reg ss class G g so e(g) = z (g G). Define parameter of infl cochar G as pair (x, E), with E G flat, x G(C), x 2 = e(e). Equivalently: pair (x, b) with b g Borel. As we saw for parameters for GL(n), Local Local : reprise param (x, E) involution w(x, E) W ; const on G (x, E); w(x, b) = rel pos of b, x b. params repns. params???

19 What parameters count Fix reg ss class G g so e(g) Z (G) (g G). Define parameter of infl cochar G = (x, E), with E G flat, x G, x 2 = e(e). Theorem parameter (x, E) 1. real form G(R) (with inv θ x = Ad(x); 2. θ x -stable T (R) G(R); 3. Borel subalgebra b t. That is: {(x, E)}/(G conj) in 1-1 corr with {(G(R), T (R), b)}/(g conj). Local Local : reprise Involution w = w(x, E) W action of θ x on T (R). Conj class of w W conj class of T (R) G(R). How many params over involution w W? Answer uses structure theory for reductive gps...

20 Counting params Max torus T G cowgt lattice X (T ) = def Hom(C, T ). Weyl group W N G (T )/T Aut(X ). Each w W has Tits representative σ w N(T ). Lie algebra t X Z C, so W acts on t. g ss /G t/w ; G has unique dom rep g t. Theorem Fix dom rep g for G, involution w W. 1. Each G orbit of params over w has rep e((g l)/2)σ w, l X s.t. (w 1)(g ρ l) = Two such reps are G-conj iff l l (w + 1)X. 3. { set of orbits over w is princ homog/ X w /(w + 1)X (w 1)(g ρ ) (w 1)X ) empty (w 1)(g ρ ) / (w 1)X ) If g X + ρ, get canonically Local Local : reprise params of infl cochar G X w /(w + 1)X.

21 Integer matrices of order 2 X lattice (Z n ), w Aut(X ), w 2 = 1. X = Z, w + = (1), X = Z, w = ( 1), X = Z 2, w s = Note: w s differs from w s by chg of basis e 1 e 1. Theorem Any w sum of copies of w +, w, w s. Z/2Z w = w + X w /(1 + w)x = 0 w = w. 0 w = w s Corollary If w = (w + ) p (w ) q (w s ) r, then rk X w = p + r rk X w = q + r dim F2 X w /(1 + w)x = p. So p, q, and r determined by w; decomp of X is not. Local Local : reprise

22 Calculations for classical groups In classical grp (GL(n), SO(2n + ɛ), Sp(2n)) X = Z n. W S n {±1} n (perm, sgn chgs of coords). Involution w partition coords {1,..., n} into 1. pos coords a i1,..., a ip, we ai = e ai ; 2. neg coords b j1,..., a jq, we bj = e bj ; 3. w s pairs (c k1, c k 1 ),..., (c kr+, c k r+ ), we ck = e c k ; and 4. w s pairs (d l1, d l 1 ),..., (d lr, d l r ), we dk = e d k. Consequently n = p + q + 2(r + + r ). So dim F2 X w /(1 + w)x = p. Example Suppose G = GL(n, C), w = ( Id) (prod of r transp): n = 0 + (n 2r) + 2(r + 0), X w /(1 + w)x = {0}. Conclude: if infl cochar g ρ + Z n, one param for each involution w; Local Local : reprise

23 Putting it all together So suppose G cplx reductive alg, G dual. Fix infl char (semisimple G orbit) H g, infl cochar (reg integral ss G orbit) G g. Definition. param (x, E) and param (y, F) said to match if w(x, E) = w(y, F) Example of matching: w(y, F) = 1 rep is principal series for split G; w(x, E) = 1 T (R) is split subgroup. Theorem. Irr reps (of infl char H) for (of infl cochar G) are in 1-1 corr with matching pairs [(x, E), (y, F)] of and params. Corollary. L-packet for param (y, F) is (empty or) princ homog space for X w /(1 w)x, w = w(y, F). Local Local : reprise

24 What did I leave out? Two cool slides called Background about arithmetic and Global conjecture discussed assembling local reps to make global rep, and when the global rep should be automorphic. Omitted two cool slides called Background about rational forms and Theorem of Kneser et al., about ratl forms of each G/Q v ratl form G/Q. Omitted interesting extensions of local results over R to study of unitary reps. Fortunately trusty sidekick Jeff Adams addresses this in 23 hours 15 minutes. Trusty sidekicks are a very Paul Sally way to get things done. Local Local : reprise

25 Local HAPPY BIRTHDAY BECKY! Local : reprise

David Vogan. MIT Lie groups seminar 9/24/14

David Vogan. MIT Lie groups seminar 9/24/14 Department of Mathematics Massachusetts Institute of Technology MIT Lie groups seminar 9/24/14 Outline Adeles Arithmetic problems matrices over Q. { ( ) } 1 0 Example: count v Z 2 t v v N. 0 1 Hard: no