Problem: Understand (classify?) the irreducible, complex - PowerPoint PPT Presentation

Parabolic induction over Z p Tyrone Crisp University of Maine Maine-Qu´ ebec Number Theory Conference 6 October 2018

Problem: Understand (classify?) the irreducible, complex representations of GL n ( Z / p ℓ Z ). � Limit as ℓ → ∞ : smooth reps of GL n ( Z p ) � ℓ = 1: solved [Frobenius, Schur, Green, Lusztig, . . . ] � ℓ > 1, n = 2 , 3: solved [Kloosterman, Kutzko, Nagornyi, . . . ] � ℓ > 1, n > 3: open, hard [Hill, Onn, Stasinski, . . . ] � Classifying irreps of GL n ( Z / p 2 Z ) for all n is wild [Nagornyi] . . . So why bother? - Decompose spaces of automorphic forms [Hecke, Kloosterman, . . . ] - Applications to other parts of rep theory [Bushnell-Kutzko, . . . ] 2 / 9

Problem: Understand (classify?) the irreducible, complex representations of GL n ( Z / p ℓ Z ). Strategy 1: induction on ℓ , via Clifford theory ( Z / p ℓ Z ։ Z / p ℓ − 1 Z ) [Shalika, Kutzko, Hill, Nagornyi, Onn, Stasinski, . . . ] Strategy 2: induction on n , via the “Philosophy of Cusp Forms” � ℓ = 1: very successful [Green, Harish-Chandra, . . . ] � ℓ > 1: work in progress with E. Meir and U. Onn � we want to be able to use both strategies together 3 / 9

Philosophy of cusp forms for GL n ( Z / p Z ) Let G n = GL n ( Z / p Z ). For each α = ( α 1 , . . . , α k ), � α i = n , consider subgroups ∗ ∗ 0       G α 1 1 α 1 G α 1 ... ... ...  ,  , L α = P α = U α =     0 0 0 1 α k G α k G α k Parabolic induction: pull back → Rep( P α ) induce i α : Rep( L α ) → Rep( G n ) − − − − − − − − − Parabolic restriction: r α : Rep( G n ) X �→ X U α − − − − − → Rep( L α ) Cusp forms: X ∈ Irr( G n ) having r α X = 0 for all proper α 4 / 9

Philosophy of cusp forms for GL n ( Z / p Z ) n = � k U α = [ 1 ∗ G n = GL n , i =1 α i , L α = [ ∗ ∗ ], P α = [ ∗ ∗ ∗ ], 1 ] i α : Rep( L α ) → Rep( G n ) r α : Rep( G n ) → Rep( L α ) cusp forms: X ∈ Irr( G n ), r α X = 0 for all proper α Theorem: [Green, Harish-Chandra] � Every X ∈ Irr( G n ) occurs as a subrep of i α ( X 1 ⊗ · · · ⊗ X k ) for some cusp forms X i ∈ Irr( G α i ) (unique up to permutations) � product of � ∼ � � � End G n i α ( X 1 ⊗ · · · ⊗ X k ) = C S m ’s Moral: Rep theory arithmetic combinatorics � = = of GL n ( Z / p Z ) (classify cuspidals) (Young tableaux) Over Z / p ℓ Z : Harder arithmetic. Same combinatorics? 5 / 9

� � � Philosophy of cusp forms for GL n ( Z / p ℓ Z )? n = � k n = GL ℓ G ℓ L ℓ P ℓ U ℓ α = [ 1 ∗ n , i =1 α i , α = [ ∗ ∗ ], α = [ ∗ ∗ ∗ ], 1 ] superscript ℓ means “over Z / p ℓ Z ”, where ℓ ≥ 1 pull back α ) induce Parabolic induction? Rep( L ℓ → Rep( P ℓ → Rep( G ℓ α ) n ) − − − − − − − − − still makes sense . . . but the resulting reps are too big. pull back to P ℓ α then induce Rep( L ℓ Rep( G ℓ α ) n ) pull back doesn’t commute! pull back � Rep( G ℓ +1 Rep( L ℓ +1 ) ) α n pull back to P ℓ +1 then induce α 6 / 9

� � � Proposal for “parabolic induction” over Z / p ℓ Z n = � k G ℓ n = GL ℓ L ℓ P ℓ U ℓ α = [ 1 ∗ n , i =1 α i , α = [ ∗ ∗ ], α = [ ∗ ∗ ∗ ], 1 ] Parabolic induction? [CMO, cf. Dat] i ℓ α : Rep( L ℓ α ) → Rep( G ℓ n ) � � � U ℓ ind G ℓ standard intertwiner ind G ℓ i ℓ α α X := Image α ) t X α X n − − − − − − − − − − − → n ( P ℓ P ℓ � For ℓ = 1, new i 1 α = old i α [Howlett-Lehrer] � i ℓ α is compatible with Clifford theory upon changing ℓ : e.g., i ℓ α Rep( L ℓ Rep( G ℓ α ) n ) commutes pull back pull back � Rep( G ℓ +1 Rep( L ℓ +1 ) ) α n i ℓ +1 α � ∃ an adjoint restriction functor r ℓ α , thus a notion of cusp forms. 7 / 9

Philosophy of cusp forms for GL n ( Z / p ℓ Z )? Conjecture: (analogue of Green’s theorem for all ℓ ≥ 1) � Every X ∈ Irr( G ℓ n ) occurs as a subrep of i ℓ α ( X 1 ⊗ · · · ⊗ X k ) for some cusp forms X i ∈ Irr( G ℓ α i ) (unique up to permutations) � product of � ∼ � i ℓ � � End G ℓ α ( X 1 ⊗ · · · ⊗ X k ) = C S m ’s n Theorem: It’s enough to verify the conjecture for nilpotent representations (with Z p replaced by a general ring of integers) . (nilpotence: Clifford-theoretic condition involving restriction to the ) ∼ minimal congruence subgroup, ker( G ℓ n ։ G ℓ − 1 = M n ( Z / p Z )) n Theorem: For α = (1 , . . . , 1): � ∼ � � product of � i ℓ End G n α ( X 1 ⊗ · · · ⊗ X n ) = C S m ’s 8 / 9

� � � Coda: equivariant homology of Bruhat-Tits buildings G : p -adic reductive group (e.g., GL n ( Q p )) Theorem: [ Higson-Nistor, Schneider, Bernstein, Keller] H G ∗ (BT( G )) : equivariant HP ∗ (Rep( G )) : periodic homology of Bruhat-Tits ∼ = cyclic homology of Rep( G ) building ≈ geometry + ≈ cohomology of Irr(G) rep thy of cmpct sbgrps Question: how does parabolic induction fit into this picture? Theorem: For G = SL 2 , L = [ ∗ ∗ ] (and perhaps more generally) : ∼ = H L ∗ (BT( L )) HP ∗ (Rep( L )) assemblage of i ℓ α s parabolic induction ∼ = � HP ∗ (Rep( G )) H G ∗ (BT( G )) 9 / 9