Matrices almost of order two Langlands philosophy Local Langlands - PowerPoint PPT Presentation

Matrices almost of order two David Vogan Matrices almost of order two Langlands philosophy Local Langlands for R David Vogan Cartan classification of real forms Department of Mathematics Local Langlands Massachusetts Institute of Technology for R : 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

Matrices almost of Outline order two David Vogan Langlands philosophy Local Langlands for R Langlands philosophy Cartan classification of real forms Local Langlands for R Local Langlands for R : reprise Cartan classification of real forms Local Langlands for R : reprise

Matrices almost of Adeles order two David Vogan Arithmetic problems � matrices over Q . Langlands philosophy � � � 1 � � 0 � v ∈ Z 2 t v Local Langlands Example: count v ≤ N . � for R 0 1 � Cartan Hard: no analysis, geometry, topology to help. classification of real forms Possible solution: use Q ֒ → R . Local Langlands � � � 1 � � for R : reprise 0 � v ∈ R 2 t v Example: find area of v ≤ N . � 0 1 � Same idea with Q ֒ → Q p leads to � ′ � ′ 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 ) .

Background about GL ( n , A ) / GL ( n , Q ) Matrices almost of order two David Vogan Langlands Gelfand: analysis re G � irr (unitary) reps of G . philosophy Local Langlands for R analysis on GL ( n , A ) / GL ( n , Q ) Cartan classification of � ′ real forms GL ( n , Q v ) � irr reps π of Local Langlands v ∈{ p , ∞} for R : reprise � ′ � GL ( n , Q v ) � π = π ( v ) , π ( v ) ∈ v ∈{ p , ∞} 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.

Matrices almost of Local Langlands order two David Vogan Big idea from Langlands unpublished 1 1973 paper: Langlands philosophy � ? GL ( n , Q v ) � n -diml reps of Gal ( Q v / Q v ) . (LLC) Local Langlands for R Big idea actually goes back at least to 1967; 1973 Cartan paper proves it for v = ∞ . classification of real forms Caveat: need to replace Gal by Weil-Deligne group. Local Langlands for R : reprise 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 . 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.

Matrices almost of Background about arithmetic order two David Vogan { Q 2 , Q 3 , . . . , Q ∞ } loc cpt fields where Q dense. Langlands If E / Q algebraic extension field, then philosophy Local Langlands for R E v = def E ⊗ Q Q v Cartan classification of is a commutative algebra over Q v . real forms Local Langlands E v is direct sum of algebraic extensions of Q v . for R : reprise 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 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 . ˇ Cebotarëv: knowing almost all σ ( v ) � σ .

Matrices almost of Global Langlands conjecture order two David Vogan Write Γ = Gal ( Q / Q ) ⊃ Gal ( Q v / Q v ) = Γ v . Langlands philosophy analysis on GL ( n , A ) / GL ( n , Q ) Local Langlands for R � ′ π GL ( n , Q ) � = 0 GL ( n , Q v ) � irr reps π of Cartan classification of v ∈{ p , ∞} real forms � ′ Local Langlands π GL ( n , Q ) � = 0 � π = π ( v ) , for R : reprise 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 ) Matrices almost of order two David Vogan All that was why it’s interesting to understand Langlands philosophy GL ( n , R ) LLC � � n -diml reps of Gal ( C / R ) Local Langlands for R � n -diml reps of Z / 2 Z Cartan � � classification of n × n cplx y , y 2 = Id / GL ( n , C ) conj real forms � Local Langlands for R : reprise 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 ) j = 1 � unitary rep π ( y ) = Ind GL ( n , R ) ξ m . B This is all irr reps of infl char zero.

Matrices almost of Integral infinitesimal characters order two David Vogan Infinitesimal char for GL ( n , R ) is unordered tuple Langlands philosophy ( γ i ∈ C ) . ( γ 1 , . . . , γ n ) , Local Langlands for R Assume first γ integral: all γ i ∈ Z . Rewrite Cartan � � classification of γ = γ 1 , . . . , γ 1 , . . . , γ r , . . . , γ r ( γ 1 > · · · > γ r ) . real forms � �� � � �� � Local Langlands m 1 terms m r terms for R : 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 | V i / 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 ) Matrices almost of order two David Vogan � � γ = γ 1 , . . . , γ 1 , . . . , γ r , . . . , γ r ( γ 1 > · · · > γ r ) ints . Langlands � �� � � �� � philosophy m 1 terms m r terms Local Langlands for R Langlands parameter of infl char γ = pair ( y , V ) , Cartan y 2 = Id , V flag, dim V i / V i − 1 = m i . classification of real forms GL ( n , R ) , infl char γ LLC � � { ( y , V} / conj by GL ( n , C ) . Local Langlands π ∈ for R : 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 | P i = + Id , y | Q i = − Id . ∼ 2. y : C ij − → C ji . 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 ) .

� � � � � � � � � � Matrices almost of Action of involution y on a flag order two David Vogan Last i rows represent subspace V i in flag. Arrows show action of y . Langlands philosophy Local Langlands P 3 Q 3 C 31 + 1 − 1 for R Cartan classification of real forms P 2 Q 2 C 21 ≃ ≃ + 1 − 1 Local Langlands V 3 for R : reprise ≃ ≃ V 2 P 1 Q 1 C 12 C 13 V 1 + 1 − 1 Represent diagram symbolically (Barbasch) � γ + 1 , . . . , γ + , γ − 1 , . . . , γ − , . . . , γ + , γ − r , . . . r , . . . , 1 1 � �� � � �� � � �� � � �� � dim P r terms dim Q r terms dim P 1 terms dim Q 1 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.

Matrices almost of General infinitesimal characters order two David Vogan Recall infl char for GL ( n , R ) is unordered tuple Langlands ( γ i ∈ C ) . philosophy ( γ 1 , . . . , γ n ) , Local Langlands Organize into congruence classes mod Z : for R Cartan � classification of γ = γ 1 , . . . , γ n 1 , γ n 1 + 1 , . . . , γ n 1 + n 2 , . . . , real forms � �� � � �� � Local Langlands cong mod Z cong mod Z for R : reprise � γ n 1 + ··· + n s − 1 + 1 , . . . , γ n , � �� � cong mod Z then in decreasing order in each congruence class: � � γ 1 1 , . . . , γ 1 , . . . , γ 1 r 1 , . . . , γ 1 , . . . , γ s 1 , . . . , γ s , . . . , γ s r s , . . . , γ s γ = 1 r 1 1 r s � �� � � �� � � �� � � �� � m s m 1 m 1 1 terms m 1 1 terms r 1 terms rs terms � �� � � �� � n s terms n 1 terms γ 1 1 > γ 1 2 > · · · > γ 1 γ s 1 > γ s 2 > · · · > γ s r 1 , · · · r s .