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

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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 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 A = A Q = R Q p = p ( ) } 1 0 v N. 0 1 v {p, } locally compact ring Q discrete subring. Q v, Arithmetic analysis on GL(n, A)/GL(n, Q).

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 ) 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.

Local Langlands Big idea from Langlands unpublished 1 1973 paper: GL(n,?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 by Harris 2 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irr rep π of GL(n, A) one n-diml rep σ(v) of Gal(Q v /Q v ) for each v. 1 Note to Impressionable Youth: big idea and unpublished go together in the career of R. Langlands. Are you R. Langlands? 2 Not that one, the other one.

Background about arithmetic {Q 2, Q 3,..., Q } loc cpt fields where Q dense. If E/Q Galois, Γ = Gal(E/Q) E v = def E Q Q v Γ is a direct sum of Galois extensions of Q v. Γ transitive on summands. Choose one summand E ν E Q Q v, define Γ v = Stab Γ (E ν ) = Gal(E ν /Q v ) Γ. Γ v Γ closed, unique up to conjugacy. E Q Q v = σ E ν σ Γ/Γ v Conclusion: n-diml σ of Γ n-diml σ(v) of Γ v. Čebotarëv: know almost all σ(v) know σ.

Global Langlands 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, } 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?

Starting local Langlands 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 Langlands: 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. This is all irr reps of infl char zero. j=1

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 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). Langlands param of infl char γ = pair (y, F) with F a flat of type γ, y n n matrix with y 2 = Id.

Integral local Langlands for GL(n, R) γ = ( ) γ 1,..., γ }{{ 1,..., γ } r,..., γ }{{} r m 1 terms m r terms (γ 1 > > γ r ) ints. Langlands parameter of infl char γ = pair (y, V), y 2 = Id, V = (V i ) flag, dim V i /V i 1 = m i. π GL(n, R), infl char γ LLC {(y, V}/conj by GL(n, C). So what are these GL(n, C) orbits? Proposition Suppose y 2 = Id n and V is a flag in C n. Have subspaces E i, A ij, B ij (i j), signs ɛ i s.t. 1. y Ei = ɛ i. 2. y : A ij B ij. 3. V i = i i E i + i i,j A i,j + j i i B j,i 4. e i = dim E i, a ij = dim A ij = dim B ij depend only on GL(n, C) (y, V).

Action of involution y on a flag Last i rows represent subspace V i in flag. Arrows show action of y. E 3 ɛ 3 B 33 A 33 B 13 B 23 V 3 E 2 ɛ 2 B 22 A 22 B 12 A 23 V 2 V 1 E 1 ɛ 1 B 11 A 11 A 12 A 13 Represent diagram symbolically (Barbasch) ( ɛ γ 1 1 }{{,.. }. ),..., γr ɛr,..., (γ 1 γ 1 ),..., (γ 1 γ 2 ),...,..., (γ r γ r ),... }{{}}{{}}{{}}{{} dim E 1 terms dim E r terms dim A 11 terms dim A 12 terms dim A r,r terms This is involution in S n plus some signs.

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.

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 }. Langlands param of infl char γ = pair (y, F) with F a flat of type γ, y n n matrix with y 2 = e(t ). )

Langlands parameters for GL(n, R) Infl char γ = (γ 1,..., γ n ) (γ i C unordered). Recall Langlands 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 Langlands parameters of infl char γ are indexed by 1. pair some (γ i, γ j ) with γ i γ j Z 0; 2. label unpaired γ k with sign ɛ k ; and 3. require ɛ j = ɛ k if γ j = γ k. 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).

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 γ g and integer k, define g(k, γ) = {X g [γ, X] = kx}. Say γ γ if γ γ + k>0 g(k, γ). Flats in g are the equivalence classes (partition each semisimple conjugacy class in g). Exponential e(γ) = exp(2πiγ) G const on flats. If G(R) split, Langlands parameter for G(R) is (y, F) with F g flat, y G, y 2 = e(f). Theorem (LLC Langlands, 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) involution w(y, F) W. G(R) possibly not split: twisted involution w(y, F).

and now for something completely different... G cplx conn red alg group. Problem: real forms of G/(equiv)? Soln (Cartan): {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.

Cartan involutions G cplx conn red alg group. cartan parameter is x G s.t. x 2 Z (G). θ x = Ad(x) Aut(G) Cartan 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 (Cartan) Surjection {cartan params} {equal rk real forms 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.

Imitating Langlands Since cartan param part of Langlands param, why not complete to a whole Langlands param? Start with z Z (G) Choose reg ss class G g so e(g) = z (g G). Define Cartan 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 Langlands parameters for GL(n), Cartan param (x, E) involution w(x, E) W ; const on G (x, E); w(x, b) = rel pos of b, x b. Langlands params repns. Cartan params???

What Cartan parameters count Fix reg ss class G g so e(g) Z (G) (g G). Define Cartan parameter of infl cochar G = (x, E), with E G flat, x G, x 2 = e(e). Theorem Cartan parameter (x, E) 1. real form G(R) (with Cartan inv θ x = Ad(x); 2. θ x -stable Cartan 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). 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 Cartan params over involution w W? Same question: # Langlands params over w W? Answer uses structure theory for reductive gps...

Counting Cartan 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 Cartan params over w has rep e((g l)/2)σ w, l X s.t. (w 1)(g ρ l) = 0. 2. 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 Cartan params of infl cochar G X w /(w + 1)X.

Integer matrices of order 2 X lattice (Z n ), w Aut(X ), w 2 = 1. 3 examples... X = Z, w + = (1), X = Z, w = ( 1), X = Z 2, w s = 0 1. 1 0 Note: w s differs from w s by chg of basis e 1 e 1. Theorem Any w Z 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.

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. Cartan param (x, E) and Langlands 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 Cartan subgroup. Theorem. Irr reps (of infl char H) for real forms (of infl cochar G) are in 1-1 corr with matching pairs [(x, E), (y, F)] of Cartan and Langlands params. Corollary. L-packet for Langlands param (y, F) is (empty or) princ homog space for X w /(1 w)x, w = w(y, F).

What did I leave out? Included cool slides called Background about arithmetic and Global Langlands conjecture discussed assembling local reps to make global rep, and when the global rep should be automorphic. Omitted two 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.

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