aa r X i v : . [ m a t h . R T ] J a n HECKE ALGEBRAS FOR TAME SUPERCUSPIDAL TYPES
KAZUMA OHARA
Abstract.
Let F be a non-archimedean local field of residue characteristic p = 2. Let G be aconnected reductive group over F that splits over a tamely ramified extension of F . In [14], Yuconstructed types which are called tame supercuspidal types and conjectured that Hecke algebrasassociated with these types are isomorphic to Hecke algebras associated with depth-zero typesof some twisted Levi subgroups of G . In this paper, we prove this conjecture. We also provethat the Hecke algebra associated with a regular supercuspidal type is isomorphic to the groupalgebra of a certain abelian group. Introduction
Let F be a non-archimedean local field of residue characteristic p = 2 and G be a connectedreductive group over F that splits over a tamely ramified extension of F . As explained in [2],the category R ( G ( F )) of smooth complex representations of G ( F ) is decomposed into a product Q [ M,σ ] G R [ M,σ ] G ( G ( F )) of full subcategories R [ M,σ ] G ( G ( F )), called Bernstein blocks . Bernsteinblocks are parametrized by inertial equivalence classes [
M, σ ] G of cuspidal pairs. Each block R [ M,σ ] G ( G ( F )) is equivalent to the category of modules over an algebra if [ M, σ ] G has an associ-ated type as explained below. Let K be a compact open subgroup of G ( F ), ( ρ, W ) be an irreduciblerepresentation of K , and s be an inertial equivalence class of a cuspidal pair. We say that ( K, ρ ) isan s -type if R s ( G ( F )) is precisely the full subcategory of R ( G ( F )) consisting of smooth represen-tations which are generated by their ρ -isotypic components. In this case, R s ( G ( F )) is equivalentto the category of modules over the Hecke algebra H ( G, ρ ) associated with (
K, ρ ) [4, Theorem 4.3].Therefore, to construct types and determine the structure of Hecke algebras associated with thetypes are essential to understand the category R ( G ( F )).In [12] and [13], Moy and Prasad defined the notion of depth of types and constructed depth-zerotypes . Hecke algebras associated with depth-zero types were calculated in [11]. In [11], Morris gavethe generators and relations for Hecke algebras associated with depth-zero types [11, Theorem 7.12].In [14], Yu constructed types of general depth which are called tame supercuspidal types . Hisconstruction starts with a tuple ( −→ G , y, −→ r , ◦ ρ − , −→ φ ), out of which it produces a sequence of types( ◦ K i , ◦ ρ i ) in G i ( F ), where −→ G = (cid:0) G ( G ( . . . ( G d = G (cid:1) is a sequence of twisted Levi subgroupsof G . Yu conjectured that the Hecke algebras associated with ( ◦ K i , ◦ ρ i ) are all isomorphic [14,Conjecture 0.2]. In particular, Hecke algebras associated with these types are isomorphic to Heckealgebras associated with depth-zero types, which are studied in [11] as explained above. In [1],Adler and Mishra proved [14, conjecture 0.2] under some conditions [1, Corollary 6.4]. However,this result covers only the cases that Hecke algebras are commutative. In this paper, we prove [14,Conjecture 0.2] without any assumptions. This is the first topic of this paper.The second topic of this paper is on regular supercuspidal types . In [10], Kaletha defined andconstructed a large class of supercuspidal representations which he calls regular . Kaletha’s con-struction starts with a regular tame elliptic pair ( S, θ ), where S is a tame elliptic maximal torus,and θ is a character of S ( F ) which satisfy some conditions [10, Definition 3.7.5]. As explainedin the paragraph following [10, Definition 3.7.3], “most” supercuspidal representations are regularwhen p is not too small. In this paper, we define and construct regular supercuspidal types , whichare constructed by the same data ( S, θ ) as Kaletha’s construction of regular supercuspidal rep-resentations. The regular supercuspidal representation constructed by (
S, θ ) contains the regularsupercuspidal type constructed by the same (
S, θ ). We prove that the Hecke algebra associatedwith the type constructed by (
S, θ ) is isomorphic to the group algebra of the quotient of S ( F ) by Mathematics Subject Classification. the unique maximal compact subgroup. This result is proved independent of [14, Conjecture 0.2]and the work of [11].We sketch the outline of this paper. In Section 3, we review Yu’s construction of supercuspidalrepresentations briefly. In Section 4, we prove [14, Conjecture 0.2]. In Section 5, we reviewKaletha’s construction of regular supercuspidal representations, define regular supercuspidal types,and determine the structure of Hecke algebras associated with regular supercuspidal types.
Acknowledgment.
I am deeply grateful to my supervisor Noriyuki Abe for his enormous supportand helpful advice. He checked the draft and gave me useful comments. I would also like to thankJessica Fintzen. She checked the previous draft and gave me a lot of comments. I am supportedby the FMSP program at Graduate School of Mathematical Sciences, the University of Tokyo.2.
Notation and assumptions
Let F be a non-archimedean local field of residue characteristic p , k F be its residue field, and G be a connected reductive group over F that splits over a tamely ramified extension of F . Wedenote by Z ( G ) the center of G and by G der the derived subgroup of G . We assume that p is anodd prime.We denote by B ( G, F ) the enlarged Bruhat–Tits building of G over F . If T is a maximal,maximally split torus of G E := G × F E for a field extension E over F , then A ( T, E ) denotes theapartment of T inside the Bruhat–Tits building B ( G E , E ) of G E over E . For any y ∈ B ( G, F ),we denote by [ y ] the projection of y on the reduced building and by G ( F ) y (resp. G ( F ) [ y ] ) thesubgroup of G ( F ) fixing y (resp. [ y ]). For y ∈ B ( G, F ) and r ∈ e R ≥ = R ≥ ∪ { r + | r ∈ R ≥ } , wewrite G ( F ) y,r for the Moy–Prasad filtration subgroup of G ( F ) of depth r [12, 13].Suppose that K is a subgroup of G ( F ) and g ∈ G ( F ). We denote gKg − by g K . If ρ is a smoothrepresentation of K , g ρ denotes the representation x ρ ( g − xg ) of g K . If Hom K ∩ g K ( g ρ, ρ ) is non-zero, we say g intertwines ρ . 3. Review of Yu’s construction
In this section, we recall Yu’s construction of supercuspidal representations and supercuspidaltypes of G ( F ) [14].An input for Yu’s construction of supercuspidal representations of G ( F ) is a tuple ( −→ G , y, −→ r , ρ − , −→ φ )where D1: −→ G = (cid:0) G ( G ( . . . ( G d = G (cid:1) is a sequence of twisted Levi subgroups of G that splitover a tamely ramified extension of F , i. e., there exists a tamely ramified extension E of F such that G iE is split for 0 ≤ i ≤ d , and (cid:0) G E ( G E ( . . . ( G dE = G E (cid:1) is a split Levisequence in G E in the sense of [14, Section 1]; we assume that Z ( G ) /Z ( G ) is anisotropic; D2: y is a point in B ( G , F ) ∩ A ( T, E ) whose projection on the reduced building of G ( F )is a vertex, where T is a maximal torus of G (hence of G i ) whose splitting field E is atamely ramified extension of F ; we denote by Φ( G i , T, E ) the corresponding root systemof G i for 0 ≤ i ≤ d ; D3: −→ r = ( r , . . . , r d ) is a sequence of real numbers satisfying ( < r < r < · · · < r d − ≤ r d ( d > , ≤ r ( d = 0); D4: ρ − is an irreducible representation of G ( F ) [ y ] such that ρ − ↾ G ( F ) y, is the inflationof a cuspidal representation of G ( F ) y, /G ( F ) y, ; D5: −→ φ = ( φ , . . . , φ d ) is a sequence of characters, where φ i is a character of G i ( F ); weassume that φ i is trivial on G i ( F ) y,r i + but non-trivial on G i ( F ) y,r i for 0 ≤ i ≤ d − r d − < r d , we assume that φ d is trivial on G d ( F ) y,r d + but non-trivial on G d ( F ) y,r d ,otherwise we assume that φ d = 1. Moreover, we assume that φ i is G i +1 -generic of depth r i relative to y in the sense of [14, Section. 9] for 0 ≤ i ≤ d − ( K i = G ( F ) [ y ] G ( F ) y,r / · · · G i ( F ) y,r i − / ,K i + = G ( F ) y, G ( F ) y, ( r / · · · G i ( F ) y, ( r i − / for 0 ≤ i ≤ d . We also define subgroups J i , J i + of G for 1 ≤ i ≤ d as follows. For α ∈ Φ( G, T, E ),let U α = U T,α denote the root subgroup of G corresponding to α . We set U = T . For x ∈ B ( G, F ), α ∈ Φ( G, T, E ) ∪ { } , and r ∈ e R ≥ , let U α ( E ) x,r denote the Moy–Prasad filtration subgroup of U α ( E ) of depth r [12, 13]. We define ( J i = G ( F ) ∩ h U α ( E ) y,r i − , U β ( E ) y,r i − / | α ∈ Φ( G i − , T, E ) ∪ { } , β ∈ Φ( G i , T, E ) \ Φ( G i − , T, E ) i ,J i + = G ( F ) ∩ h U α ( E ) y,r i − , U β ( E ) y, ( r i − / | α ∈ Φ( G i − , T, E ) ∪ { } , β ∈ Φ( G i , T, E ) \ Φ( G i − , T, E ) i for 1 ≤ i ≤ d . As explained in [14, Section 1], J i and J i + are independent of the choice of a maximaltorus T of G so that T splits over a tamely ramified extension E of F and y ∈ A ( T, E ).Yu constructed irreducible representations ρ i and ρ ′ i of K i for 0 ≤ i ≤ d inductively. First, weput ρ ′ = ρ − , ρ = ρ ′ ⊗ φ .Suppose that ρ i − and ρ ′ i − are already constructed, and ρ ′ i − ↾ G i − ( G ) y,ri − is 1-isotypic. In[14, Section 11], Yu defined a representation φ ′ i − of K i using the theory of Weil representation.This representation only depends on φ i − . If r i − < r i , φ ′ i − ↾ G i ( F ) y,ri is 1-isotypic. Let inf (cid:0) ρ ′ i − (cid:1) be the inflation of ρ ′ i − via the map K i = K i − J i → K i − J i /J i ≃ K i − /G i − ( G ) y,r i − . Nowwe define ρ ′ i = inf (cid:0) ρ ′ i − (cid:1) ⊗ φ ′ i − , which is trivial on G i ( F ) y,r i if r i − < r i . Finally, we define ρ i = ρ ′ i ⊗ φ i .We explain the construction of φ ′ i − . Let h· , ·i i be a pairing on J i /J i + defined by h a, b i i =ˆ φ i − ( aba − b − ). Here, ˆ φ i − denotes an extension of φ i − ↾ K G i − ( F ) y, to K G i − ( F ) y, G ( F ) y, ( r i − / defined in [14, Section 4]. The pairing is well-defined because by [5, Proposition 6.4.44], [ J i , J i ]is contained in J i + . Note that since the order of every element in J i /J i + divides p , we can re-gard J i /J i + as a F p -vector space. By [14, Lemma 11.1], this pairing is non-degenerate on J i /J i + .In addition, by the construction of ˆ φ i − , for j ∈ J i + , j p is contained in Ker( ˆ φ i − ). Therefore,the order of every element in ˆ φ i − ( J i + ) divides p , and since ˆ φ i − ( J i + ) is a non-trivial subgroupof C × , this implies that ˆ φ i − ( J i + ) is isomorphic to F p . Hence we can regard h· , ·i i as a non-degenerate F p -valued pairing on J i /J i + and J i /J i + as a symplectic space over F p . For a sym-plectic space ( V, h , i ) over F p , we define the Heisenberg group V of V to be the set V × F p with the group law ( v, a )( w, b ) = ( v + w, a + b + h v, w i ). Yu constructed a canonical isomorphism j : J i / (cid:16) J i + ∩ Ker( ˆ φ i − ) (cid:17) → ( J i /J i + ) × J i + / (cid:16) J i + ∩ Ker( ˆ φ i − ) (cid:17) ≃ ( J i /J i + ) in [14, Proposition 11.4].Combining this isomorphism and the map K i − → Sp (cid:0) J i /J i + (cid:1) induced by the conjugation, wedefine K i − ⋉ J i → K i − ⋉ (cid:16) J i / (cid:16) J i + ∩ Ker( ˆ φ i − ) (cid:17)(cid:17) → Sp (cid:0) J i /J i + (cid:1) ⋉ ( J i /J i + ) . Combining thismap and the Weil representation of Sp (cid:0) J i /J i + (cid:1) ⋉ ( J i /J i + ) associated with the central characterˆ φ i − , we construct a representation ˜ φ i − of K i − ⋉ J i . Let inf( φ i − ) be the inflation of φ i − via themap K i − ⋉ J i → K i − , then inf( φ i − ) ⊗ ˜ φ i − factors through the map K i − ⋉ J i → K i − J i = K i .We define φ ′ i − be the representation of K i whose inflation to K i − ⋉ J i is inf( φ i − ) ⊗ ˜ φ i − .For open subgroups K , K of G i ( F ) and a representation ρ of K , let ind K K ρ denote thecompactly induced representation of K . Theorem 3.1 ([14, Theorem 15.1]) . The representation π i = ind G i ( F ) K i ρ i of G i ( F ) is an irreduciblesupercuspidal representation of depth r i for ≤ i ≤ d . For the proof of this theorem, Yu uses [14, Proposition 14.1] and [14, Theorem 14.2], which arepointed out in [8] to be false. However, Fintzen uses an alternative approach in [8] and proves [14,Theorem 15.1] without using [14, Proposition 14.1] or [14, Theorem 14.2].Next, we review Yu’s construction of supercuspidal types. We start with a datum ( −→ G , y, −→ r , ◦ ρ − , −→ φ )satisfying D1 , D2 , D3 , D5 and ◦ D4 : ◦ D4: ◦ ρ − is an irreducible representation of G ( F ) y such that ◦ ρ − ↾ G ( F ) y, is the inflationof a cuspidal representation of G ( F ) y, /G ( F ) y, ,instead of D4 . We then follow the above construction replacing K i with ◦ K i = G ( F ) y G ( F ) y,r / · · · G i ( F ) y,r i − / , and construct an irreducible representation ◦ ρ i of ◦ K i . KAZUMA OHARA
Proposition 3.2.
Let ( −→ G , y, −→ r , ρ − , −→ φ ) be a datum satisfying D1 , D2 , D3 , D4 , D5 and ◦ ρ − bean irreducible representation of G ( F ) y which is contained in ρ − ↾ G ( F ) y . Then ( −→ G , y, −→ r , ◦ ρ − , −→ φ ) satisfies D1 , D2 , D3 , ◦ D4 , D5 and the representation ◦ ρ i constructed above is an s i - type in thesense of [4] , where s i is the inertial equivalence class of [ G i , π i ] G i .Proof. Note that ◦ K i is the unique maximal compact subgroup of K i , and ◦ ρ i is contained in ρ i ↾ ◦ K i . Then this proposition follows from [4, Proposition 5.4]. (cid:3) Types obtained in this way are called tame supercuspidal types . In the following, we write K d , K d + , ◦ K d , ρ d , ◦ ρ d , π d simply by K, K + , ◦ K, ρ, ◦ ρ, π , respectively.4. Hecke algebra isomorphism
Let K be a compact open subgroup of G ( F ) and ( ρ, W ) be an irreducible representation of K . We define a Hecke algebra H ( G ( F ) , ρ ) associated with ( K, ρ ) as in [4, Section 2] and writeˇ H ( G ( F ) , ρ ) for H ( G ( F ) , ˇ ρ ), where ˇ ρ is the contragradient of ρ . So, ˇ H ( G ( F ) , ρ ) is the C -vectorspace of compactly supported functions Φ : G ( F ) → End C ( W ) satisfyingΦ( k gk ) = ρ ( k ) ◦ Φ( g ) ◦ ρ ( k ) , k i ∈ K, g ∈ G ( F ) , and for Φ , Φ ∈ ˇ H ( G ( F ) , ρ ), the product Φ ∗ Φ is defined by(Φ ∗ Φ ) ( x ) = Z G ( F ) Φ ( y ) ◦ Φ ( y − x ) dy. Here, we normalize the Haar measure of G ( F ) so that vol( K ) = 1. If s is an inertial equivalenceclass of a cuspidal pair and ( K, ρ ) is an s -type, the Bernstein block associated with s is equivalent tothe category of H ( G ( F ) , ρ )-modules [4, Theorem 4.3]. We restrict our attention to Hecke algebrasassociated with tame supercuspidal types, defined in Section 3.Let ( −→ G , y, −→ r , ρ − , −→ φ ) be a datum satisfying D1 , D2 , D3 , D4 , D5 and ◦ ρ − be an irreduciblerepresentation of G ( F ) y which is contained in ρ − ↾ G ( F ) y . We construct a [ G, π ] G -type ( ◦ K, ◦ ρ )as in Section 3.We define supp (cid:0) ˇ H ( G ( F ) , ◦ ρ ) (cid:1) = [ f ∈ ˇ H ( G ( F ) , ◦ ρ ) supp( f ) , where supp( f ) denotes the support of f . We call it the support of ˇ H ( G ( F ) , ◦ ρ ). Note thatsupp (cid:0) ˇ H ( G ( F ) , ◦ ρ ) (cid:1) = { g ∈ G ( F ) | g intertwines ◦ ρ } . Proposition 4.1.
The support of ˇ H ( G ( F ) , ◦ ρ ) is contained in ◦ KG ( F ) [ y ] ◦ K . Moreover, an ele-ment g ∈ G ( F ) [ y ] intertwines ◦ ρ if and only if g intertwines ◦ ρ − . Remark 4.2.
By [14, Remark 3.5], G ( F ) [ y ] normalizes ◦ K , so ◦ KG ( F ) [ y ] ◦ K = G ( F ) [ y ] ◦ K .Proposition 4.1 follows easily from [14, Corollary 15.5]. However [14, Corollary 15.5] relies on[14, Proposition 14.1] and [14, Theorem 14.2], which are pointed out in [8] to be false. In thefollowing, we prove Proposition 4.1 using an argument by Fintzen in [8, Theorem 3.1], which doesnot rely on [14, Proposition 14.1] or [14, Theorem 14.2].For the first claim, it is enough to show that if g ∈ G ( F ) intertwines ◦ ρ , then g ∈ ◦ KG ( F ) [ y ] ◦ K .The first step is the following Lemma. Lemma 4.3. If g ∈ G ( F ) intertwines ◦ ρ , then g ∈ ◦ KG ( F ) ◦ K .Proof. This follows from [14, Proposition 4.1] and [14, Proposition 4.4]. Note that [14, Proposi-tion 4.4] is also true if we replace ρ i with ◦ ρ . (cid:3) Next, we prove the following Lemma.
Lemma 4.4. If g ∈ G ( F ) intertwines ◦ ρ , then g ∈ G ( F ) [ y ] . ECKE ALGEBRAS FOR TAME SUPERCUSPIDAL TYPES 5
Proof.
Let f be a nonzero element of Hom ◦ K ∩ g ( ◦ K ) ( g ( ◦ ρ ) , ◦ ρ ). We write V f for the image of f . By[14, Proposition 4.4], ◦ ρ ↾ K + is θ d -isotypic, where θ d = d Y j =0 ˆ φ j ↾ K + . This implies that G ( F ) y, ( ⊂ K + ) acts on ◦ ρ by θ d , and g G ( F ) y, acts on g ( ◦ ρ ) by g θ d . For h ∈ G ( F ) y, ∩ g G ( F ) y, and 0 ≤ j ≤ d , g ˆ φ j ( h ) = ˆ φ j ( g − hg ) = φ j ( g − hg ) = φ j ( g ) − φ j ( h ) φ j ( g ) = φ j ( h ) = ˆ φ j ( h ) . Hence G ( F ) y, ∩ g G ( F ) y, acts on g ( ◦ ρ ) by θ d . Therefore, if we let U ′− = (cid:0) G ( F ) y, ∩ g G ( F ) y, (cid:1) G ( F ) y, ,U ′− acts on V f by θ d .From the construction, ◦ ρ is decomposed as ◦ ρ = d O i = − V i , where V − is the inflation of ◦ ρ − via the map ◦ K = G ( F ) y J · · · J d → G ( F ) y J · · · J d /J · · · J d ≃ G ( F ) y /G ( F ) y,r ,V i is the inflation of φ ′ i via the map ◦ K = ◦ K i +1 J i +2 · · · J d → ◦ K i +1 J i +2 · · · J d /J i +2 · · · J d = ◦ K i +1 /G i +1 ( F ) y,r i +1 , for 0 ≤ i ≤ d − V d − = φ ′ d − , and V d = φ d .Let T be a maximal torus of G so that T splits over a tamely ramified extension E of F and y, g · y ∈ A ( T, E ). Such a torus exists by the fact that any two points of B ( G , F ) is contained inan apartment of B ( G , F ) and by the discussion in the beginning of [14, Section 2]. We define U i = G ( F ) ∩ h U α ( E ) y,r i / | α ∈ Φ( G i +1 , T, E ) \ Φ( G i , T, E ) , α ( y − g · y ) < i , for 0 ≤ i ≤ d −
1. Since U i is contained in J i +1 , the action of U i on V j is trivial for − ≤ j ≤ i − i + 1 ≤ j , U i is contained in ◦ K j , which acts on V j by φ ′ j ↾ ◦ K j = φ j ⊗ ˜ φ j . Here, the action of ◦ K j by ˜ φ j factors through the map ◦ K j → Sp (cid:16) J j +1 /J j +1+ (cid:17) induced by the conjugation. By [5,Proposition 6.4.44], [ J j +1 , U i ] ⊂ [ J j +1 , G j ( F ) y, ] ⊂ J j +1+ . Therefore, the action of U i by ˜ φ j is trivial, hence U i acts on V j by φ j . For j = i , U i acts on V i bythe Heisenberg representation ˜ φ i of J i +1 /J i +1+ . Putting these arguments together, we see that theaction of U i by ◦ ρ is (cid:0) ⊗ i − j = − Id V j (cid:1) ⊗ ˜ φ i ⊗ (cid:0) ⊗ dj = i +1 φ j (cid:1) . On the other hand, for α ∈ Φ( G i +1 , T, E ) \ Φ( G i , T, E ) which satisfies α ( y − g · y ) <
0, we have g − U α ( E ) y,r i / = g − U α ( E ) g · y, ( r i / − α ( y − g · y ) = U g − α ( E ) y, ( r i / − α ( y − g · y ) ⊂ U g − α ( E ) y, ( r i / . Here, g − α denotes the character t α ( gtg − ) of g − T , and U g − α = U g − T ,g − α denotes thecorresponding root subgroup. Since J i +1+ is independent of the choice of a maximal torus, and g − T is a maximal torus of G so that g − T splits over E and y ∈ A ( g − T, E ), g − U i ⊂ J i +1+ ⊂ K i +1+ ⊂ K d + . As ◦ ρ ↾ K d + is θ d -isotypic, U i acts on V f by g θ d ↾ U i .By the construction of ˆ φ j in [14, Section 4], ˆ φ j is trivial on G ( F ) ∩ h U g − α ( E ) y, ( r j / | α ∈ Φ( G, T, E ) \ Φ( G j , T, E ) i KAZUMA OHARA for 0 ≤ j ≤ d −
1. Therefore, for j ≤ i , g − U i is contained in Ker( ˆ φ j ). This implies that g θ d ↾ U i = d Y j = i +1 g ˆ φ j ↾ U i = d Y j = i +1 g φ j ↾ U i = d Y j = i +1 φ j ↾ U i . Hence U i acts on V f by (cid:0) ⊗ ij = − Id V j (cid:1) ⊗ (cid:0) ⊗ dj = i +1 φ j (cid:1) . Comparing the action of U i by ◦ ρ and the action of U i on V f , we conclude that V f is contained in V − ⊗ ( ⊗ d − i =0 V U i i ) ⊗ V d .We study the subspace V U i i for 0 ≤ i ≤ d −
1. Recall that V i is the space of the Weil representationof Sp (cid:0) J i +1 /J i +1+ (cid:1) ⋉ ( J i +1 /J i +1+ ) . We write W i +1 = J i +1 /J i +1+ . We define the subspace ( W i +1 ) to be the image of U i in W i +1 , ( W i +1 ) to be the image of G ( F ) ∩ h U α ( E ) y,r i / | α ∈ Φ( G i +1 , T, E ) \ Φ( G i , T, E ) , α ( y − g · y ) = 0 i in W i +1 , and ( W i +1 ) to be the image of G ( F ) ∩ h U α ( E ) y,r i / | α ∈ Φ( G i +1 , T, E ) \ Φ( G i , T, E ) , α ( y − g · y ) > i in W i +1 . Note that ( W i +1 ) k is written by V k in [14, Section 13] for 1 ≤ k ≤
3. By [14, Lemma 13.6],( W i +1 ) , ( W i +1 ) are totally isotropic subspaces of the symplectic space (cid:0) W i +1 , h , i i +1 (cid:1) and( W i +1 ) ⊥ = ( W i +1 ) ⊕ ( W i +1 ) , ( W i +1 ) ⊥ = ( W i +1 ) ⊕ ( W i +1 ) . Let P i +1 be the maximal parabolic subgroup of Sp( W i +1 ) that preserves ( W i +1 ) . Then we obtainthe natural map ι : P i +1 → Sp (cid:16) ( W i +1 ) ⊥ / ( W i +1 ) (cid:17) ≃ Sp (cid:0) ( W i +1 ) (cid:1) . We write ( ˜ φ i ) for the Weil representation of Sp (cid:0) ( W i +1 ) (cid:1) ⋉ (( W i +1 ) ) associated with thecentral character ˆ φ i . By [9, Theorem 2.4 (b)], the restriction of ˜ φ i from Sp (cid:0) W i +1 (cid:1) ⋉ ( W i +1 ) to P i +1 ⋉ ( W i +1 ) is given byind P i +1 ⋉ ( W i +1 ) P i +1 ⋉ (( W i +1 ) ⊕ (( W i +1 ) ) ) ( ˜ φ i ) ⊗ ( χ i +1 ⋉ . Here, we regard ( ˜ φ i ) be a representation of P i +1 ⋉ (cid:0) ( W i +1 ) ⊕ (( W i +1 ) ) (cid:1) by defining the actionof ( W i +1 ) to be trivial and defining the action of P i +1 to be the composition of ι and ( ˜ φ i ) . Thecharacter χ i +1 of P i +1 is χ E + of [9, Lemma 2.3 (d)], which factors through the natural map P i +1 → GL(( W i +1 ) ). Since ( W i +1 ) is a complete system of representatives for (cid:0) P i +1 ⋉ ( W i +1 ) (cid:1) / (cid:0) P i +1 ⋉ (cid:0) ( W i +1 ) ⊕ (( W i +1 ) ) (cid:1)(cid:1) , as a representation of P i +1 ⋉ (cid:0) ( W i +1 ) ⊕ (( W i +1 ) ) (cid:1) ,ind P i +1 ⋉ ( W i +1 ) P i +1 ⋉ (( W i +1 ) ⊕ (( W i +1 ) ) ) ( ˜ φ i ) ⊗ ( χ i +1 ⋉ ≃ M v ∈ ( W i +1 ) v ( ˜ φ i ) ⊗ ( χ i +1 ⋉ . Since ( W i +1 ) acts on ( ˜ φ i ) trivially, ( W i +1 ) acts v ( ˜ φ i ) by v ˆ φ i ( v − v v v − ) = h v − , v i i +1 . We note that ( W i +1 ) ⊥ = ( W i +1 ) ⊕ ( W i +1 ) . Hence for every element v ∈ ( W i +1 ) there exists v ∈ ( W i +1 ) such that h v − , v i i +1 = 0. Therefore, (cid:16) ind P i +1 ⋉ ( W i +1 ) P i +1 ⋉ ( W i +1 ) ⊕ (( W i +1 ) ) ( ˜ φ i ) ⊗ ( χ i +1 ⋉ (cid:17) { } ⋉ (( W i +1 ) ×{ } ) ≃ ( ˜ φ i ) ⊗ χ i +1 as a representation of P i +1 .The image of U i via the special isomorphism constructed in [14, Proposition 11.4] is ( W i +1 ) ×{ } ⊂ ( W i +1 ) . Therefore P i +1 acts on V U i i by ( ˜ φ i ) ⊗ χ i +1 .We define U − = G ( F ) ∩ h U α ( E ) y, | α ∈ Φ( G , T, E ) , α ( y − g · y ) < i . Then, U ′− is contained in U − G ( F ) y, . By [5, Proposition 6.4.44],[ J i +1 , G ( F ) y, ] ⊂ J i +1+ , ECKE ALGEBRAS FOR TAME SUPERCUSPIDAL TYPES 7 so the image of G ( F ) y, in Sp (cid:0) W i +1 (cid:1) is trivial. Also, by [5, Proposition 6.4.44],[ G i +1 ( F ) y,f (1 , , U − ] ⊂ G i +1 ( F ) y,f , where f (1 ,
2) and f are functions on Φ( G i +1 , T, E ) ∪ { } defined by f (1 , ( α ) = r i (cid:0) α ∈ Φ( G i , T, E ) ∪ { } (cid:1) r i (cid:0) α ∈ Φ( G i +1 , T, E ) \ Φ( G i , T, E ) , α ( y − g · y ) ≤ (cid:1) ( r i )+ (cid:0) α ∈ Φ( G i +1 , T, E ) \ Φ( G i , T, E ) , α ( y − g · y ) > (cid:1) ,f ( α ) = r i (cid:0) α ∈ Φ( G i , T, E ) ∪ { } (cid:1) r i (cid:0) α ∈ Φ( G i +1 , T, E ) \ Φ( G i , T, E ) , α ( y − g · y ) < (cid:1) ( r i )+ (cid:0) α ∈ Φ( G i +1 , T, E ) \ Φ( G i , T, E ) , α ( y − g · y ) ≥ (cid:1) and for h = f (1 , , f , G i +1 ( F ) y,h = G ( F ) ∩ h U α ( E ) y,h ( α ) | α ∈ Φ( G i +1 , T, E ) ∪ { }i . Note that the image of G i +1 ( F ) y,f (1 , (resp. G i +1 ( F ) y,f ) in W i +1 is ( W i +1 ) ⊕ ( W i +1 ) (resp.( W i +1 ) ). Therefore, the image of U − in Sp (cid:0) W i +1 (cid:1) is contained in P i +1 , and the image of U − via the map ι : P i +1 → Sp (cid:16) ( W i +1 ) ⊥ / ( W i +1 ) (cid:17) ≃ Sp (cid:0) ( W i +1 ) (cid:1) is trivial. Moreover, since U − is a prp- p subgroup of G ( F ), the image in GL(( W i +1 ) ) under thenatural map P i +1 → GL(( W i +1 ) ) is a p -group, hence contained in the commutator subgroup ofGL(( W i +1 ) ). Therefore, χ i +1 is trivial on the image of U − . These arguments imply that ˜ φ i ( U ′− )is trivial for 0 ≤ i ≤ d −
1, so the action of U ′− on ( ⊗ d − i =0 V U i i ) ⊗ V d is θ d -isotypic. Since U ′− acts on V f by θ d and V f is contained in V − ⊗ ( ⊗ d − i =0 V U i i ) ⊗ V d , this implies that V U ′− − is nonzero. As [ y ] is avertex, if g G ( F ) [ y ] , the image of U ′− in G ( F ) y, /G ( F ) y, is a unipotent radical of a properparabolic subgroup of G ( F ) y, /G ( F ) y, . This contradicts that ◦ ρ − ↾ G ( F ) y, is the inflation ofa cuspidal representation of G ( F ) y, /G ( F ) y, . Therefore we obtain that g ∈ G ( F ) [ y ] . (cid:3) We prove the second claim of Proposition 4.1 using an argument in [14, Proposition 4.6]. Since G ( F ) [ y ] is contained in K and V i is a restriction of a representation of K to ◦ K for 0 ≤ i ≤ d , if g ∈ G ( F ) [ y ] intertwines ◦ ρ − , then g intertwines ◦ ρ . We prove that the converse is true. Assumethat g ∈ G ( F ) [ y ] intertwines ◦ ρ . Since V d is a restriction of a character of G ( F ), g also intertwines ⊗ d − j = − V i . If d = 0 it implies that g intertwines ◦ ρ − . Suppose d ≥
1. We prove that g intertwines ⊗ d − j = − V i . Let f be a nonzero element of Hom ◦ K ( g ( ◦ ρ ) , ◦ ρ ). We write f = P j f ′ j ⊗ f ′′ j , where f ′ j ∈ End C ( ⊗ d − i = − V i ) and f ′′ j ∈ End C ( V d − ). We may assume that { f ′ j } is a linearly independentset.Since the action of J d on ⊗ d − i = − V i is trivial, for x ∈ J d we obtain X j f ′ j ⊗ (cid:0) f ′′ j ◦ ( g φ ′ d − )( x ) (cid:1) = X j f ′ j ⊗ (cid:0) φ ′ d − ( x ) ◦ f ′′ j (cid:1) . The linearly independence of { f j } implies that f ′′ j ∈ Hom J d ( g φ ′ d − , φ ′ d − ). By [14, Proposi-tion 12.3], Hom J d ( g φ ′ d − , φ ′ d − ) is 1-dimensional, so we may assume that there is only one j .We write f = f ′ ⊗ f ′′ , where f ′ ∈ End C ( ⊗ d − i = − V i ) and f ′′ ∈ Hom J d ( g φ ′ d − , φ ′ d − ). Since g ∈ G ( F ) [ y ] ⊂ K , Hom K ( g φ ′ d − , φ ′ d − ) = End K ( φ ′ d − ) is 1-dimensional, and it is a subspace ofHom J d ( g φ ′ d − , φ ′ d − ), which is also 1-dimensional. Therefore Hom J d ( g φ ′ d − , φ ′ d − ) = Hom K ( g φ ′ d − , φ ′ d − ),and f ′′ is K -equivariant. This implies that f ′ is a nonzero element in Hom ◦ K ( g ( ⊗ d − j = − V i ) , ⊗ d − j = − V i ),and g intertwines ⊗ d − j = − V i . Then, an inducting argument implies that g intertwines ◦ ρ − .We now prove [14, Conjecture 0.2]. Theorem 4.5.
There is a support-preserving algebra isomorphism ˇ H ( G ( F ) , ◦ ρ ) ≃ ˇ H (cid:0) G ( F ) , ◦ ρ − (cid:1) . Here, we say that an isomorphism η : ˇ H ( G ( F ) , ◦ ρ ) → ˇ H (cid:0) G ( F ) , ◦ ρ − (cid:1) is support-preserving iffor every f ∈ ˇ H ( G ( F ) , ◦ ρ ), supp( f ) = ◦ K supp( η ( f )) ◦ K . KAZUMA OHARA
Proof.
We set G ◦ ρ = { g ∈ G ( F ) [ y ] | g intertwines ◦ ρ } = { g ∈ G ( F ) [ y ] | g intertwines ◦ ρ − } . The second equation follows from Proposition 4.1. By Proposition 4.1 and Remark 4.2, the supportof ˇ H ( G ( F ) , ◦ ρ ) is G ◦ ρ ◦ K , and the support of ˇ H (cid:0) G ( F ) , ◦ ρ − (cid:1) is G ◦ ρ ◦ K . Fix a complete system ofrepresentatives { g i } i ⊂ G ◦ ρ for G ◦ ρ ◦ K/ ◦ K = G ◦ ρ ◦ K / ◦ K = G ◦ ρ / (cid:0) G ◦ ρ ∩ ◦ K (cid:1) . For each g i , Hom ◦ K ( g i ( ◦ ρ − ) , ◦ ρ − ) is 1-dimensional. We fix a basis ( T g i ) − of this space. Wedefine T g i = ( T g i ) − ⊗ φ ′ ( g i ) ⊗ . . . ⊗ φ ′ d ( g i ) . The element T g i is a basis of the 1-dimensional vector space Hom ◦ K ( g i ( ◦ ρ ) , ◦ ρ ). We define f g i ∈ ˇ H ( G ( F ) , ◦ ρ ) and ( f g i ) − ∈ ˇ H (cid:0) G ( F ) , ◦ ρ − (cid:1) by f g i ( x ) = ( T g i ◦ ◦ ρ ( k ) ( x = g i k, k ∈ ◦ K )0 (otherwise) , ( f g i ) − ( x ) = ( ( T g i ) − ◦ ◦ ρ − ( k ) ( x = g i k, k ∈ ◦ K )0 (otherwise) . Then as vector spaces,ˇ H ( G ( F ) , ◦ ρ ) = M i C f g i , ˇ H (cid:0) G ( F ) , ◦ ρ − (cid:1) = M i C ( f g i ) − and f g i ( f g i ) − gives a support preserving vector space isomorphism ˇ H ( G ( F ) , ◦ ρ ) → ˇ H (cid:0) G ( F ) , ◦ ρ − (cid:1) .We will show that it is an algebra isomorphism. Let g i , g i be representatives, and take g i sothat g i g i ∈ g i ◦ K . We simply write g , g , g for g i , g i , g i , respectively. For x ∈ G , we have( f g ∗ f g )( x ) = Z G f g ( y ) ◦ f g ( y − x ) dy = Z ◦ K T g ◦ ◦ ρ ( k ) ◦ f g ( k − g − x ) dk = ( T g ◦ T g ◦ ◦ ρ (( g g ) − g k ′ ) ( x = g k ′ , k ′ ∈ ◦ K )0 (otherwise)= c · f g ( x ) , where c satisfies c · T g = T g ◦ T g ◦ ◦ ρ (( g g ) − g ) . By a similar calculation, we obtain( f g ) − ∗ ( f g ) − = c − · ( f g ) − , where c − satisfies c − · ( T g ) − = ( T g ) − ◦ ( T g ) − ◦ ◦ ρ − (( g g ) − g ) . By the definition of T g i , T g ◦ T g ◦ ◦ ρ (( g g ) − g )= (cid:0) ( T g ) − ◦ ( T g ) − ◦ ◦ ρ − (( g g ) − g ) (cid:1) ⊗ d O j =0 φ ′ j ( g ) ◦ φ ′ j ( g ) ◦ φ ′ j (( g g ) − g ) = (cid:0) ( T g ) − ◦ ( T g ) − ◦ ◦ ρ − (( g g ) − g ) (cid:1) ⊗ d O j =0 φ ′ j ( g ) = c − · ( T g ) − ⊗ d O j =0 φ ′ j ( g ) = c − · T g . ECKE ALGEBRAS FOR TAME SUPERCUSPIDAL TYPES 9
This implies that c = c − and the isomorphism constructed above is an algebra isomorphism. (cid:3) Hecke algebras for regular supercuspidal types
Firstly, we review the definition and the construction of regular supercuspidal representations.In this section, we assume that p is odd, is not a bad prime for G , and dose not divide the orderof the fundamental group of G der . These assumptions are needed for the existence of a Howefactorization.Let ( S, θ ) be a regular tame elliptic pair, i. e., S is a tame elliptic maximal torus of G , and θ is a character of S ( F ) which satisfy the conditions in [10, Definition 3.7.5]. Let [ y ] be the pointof reduced building of G over F which is associated to S in the sense of the paragraph above [10,Lemma 3.4.3] and chose y ∈ B ( G, F ) such that the projection of y on the reduced building is [ y ].From ( S, θ ), we define a sequence of twisted Levi subgroups −→ G = (cid:0) S = G − ⊂ G ( . . . ( G d = G (cid:1) in G and a sequence of real numbers −→ r = (0 = r − , r , . . . , r d ) as in [10, 3.6].A Howe factorization of ( S, θ ) is a sequences of characters −→ φ = ( φ − , . . . , φ d ), where φ i is acharacter of G i ( F ) satisfying θ = d Y i = − φ i ↾ S ( F ) and some additional technical conditions (see [10, Definition 3.6.2]). By [10, Proposition 3.6.7],( S, θ ) has a Howe factorization. We take a Howe factorization −→ φ . Using the pair ( S, φ − ), wedefine an irreducible representation ρ − of G ( F ) [ y ] as follows.Let G ◦ y be the reductive quotient of the special fiber of the connected parahoric group schemeof G associated to y and S ◦ be the reductive quotient of the special fiber of the connected N´eronmodel of S . Then S ◦ ⊂ G ◦ y is an elliptic maximal torus. The restriction of φ − to S ( F ) factorsthrough a character ¯ φ − : S ◦ ( k F ) → C × .Let κ ( S, ¯ φ − ) = ± R S ◦ , ¯ φ − be the irreducible cuspidal representation of G ◦ y ( k F ) arising from theDeligne–Lusztig construction applied to S ◦ and ¯ φ − [6, Section 1]. We identify it with its inflationto G ( F ) y, . We can extend κ ( S, ¯ φ − ) to a representation κ ( S,φ − ) of S ( F ) G ( F ) y, [10, 3.4.4]. Nowwe define ρ − = ind G ( F ) [ y ] S ( F ) G ( F ) y, κ ( S,φ − ) . Proposition 5.1.
Let ( S, θ ) be a regular tame elliptic pair and G i , r i , φ i , ρ − be as above. Then, (cid:0) ( G i ) di =0 , y, ( r i ) di =0 , ρ − , ( φ i ) di =0 (cid:1) satisfies D1 , D2 , D3 , D4 , D5 .Proof. This is a part of [10, Proposition 3.7.8]. (cid:3)
Using this datum, we construct an irreducible supercuspidal representation π ( S,θ ) of G ( F ), whichonly depends on ( S, θ ). An irreducible supercuspidal representation of G ( F ) obtained in this wayis called regular .Next, we define regular supercuspidal types . Let ( S, θ ) be a regular tame elliptic pair and define G i , r i , φ i , κ ( S,φ − ) as above. We define ◦ ρ − = ind G ( F ) y G ( F ) y ∩ S ( F ) G ( F ) y, ◦ κ ( S,φ − ) , where ◦ κ ( S,φ − ) denotes the restriction of κ ( S,φ − ) to G ( F ) y ∩ S ( F ) G ( F ) y, . Proposition 5.2.
The representation ◦ ρ − is an irreducible representation of G ( F ) y .Proof. This is essentially the same as [10, Proposition 3.4.20], which is the same as the one for [7,Lemma 4.5.1].Since G ( F ) y ∩ S ( F ) G ( F ) y, contains G ( F ) y, and κ ( S,φ − ) ↾ G ( F ) y, = κ ( S, ¯ φ − ) is irreducible, ◦ κ ( S,φ − ) is also irreducible. Therefore it is enough to show that if g ∈ G ( F ) y intertwines ◦ κ ( S,φ − ) ,then g ∈ G ( F ) y ∩ S ( F ) G ( F ) y, . Suppose g ∈ G ( F ) y intertwines ◦ κ ( S,φ − ) , then g also intertwines κ ( S, ¯ φ − ) . Hence by [6, Theorem 6.8], there is h ∈ G ( F ) y, so that Ad( hg ) (cid:0) S ◦ , ¯ φ − (cid:1) = (cid:0) S ◦ , ¯ φ − (cid:1) .By [10, Lemma 3.4.5], there is l ∈ G ( F ) y, so that Ad( lhg ) (cid:0) S, φ − ↾ S ( F ) (cid:1) = (cid:0) S, φ − ↾ S ( F ) (cid:1) .Thus lhg ∈ S ( F ) by the regularity of θ , and g ∈ G ( F ) y ∩ S ( F ) G ( F ) y, . (cid:3) This proposition implies that (cid:0) ( G i ) di =0 , y, ( r i ) di =0 , ◦ ρ − , ( φ i ) di =0 (cid:1) satisfies D1 , D2 , D3 , ◦ D4 , D5 .We construct a [ G, π ( S,θ ) ] G -type ( ◦ K, ◦ ρ ) from this datum. We call types obtained in this way regular supercuspidal types . Remark 5.3.
Let (
S, θ ) be a regular tame elliptic pair. Let ρ − be the representation of G ( F ) [ y ] and ◦ ρ − be the representation of G ( F ) y defined as above. If ρ ′ is an irreducible representationof G ( F ) y which is contained in ρ − ↾ G ( F ) y then there is g ∈ G ( F ) [ y ] so that ρ ′ is equivalent to g ( ◦ ρ − ). Now, ρ ′ ≃ g ( ◦ ρ − ) ≃ ind G ( F ) y G ( F ) y ∩ gS ( F ) g − G ( F ) y, ◦ κ ( gSg − , g φ ) and (cid:0) gSg − , g φ (cid:1) is a regular tame elliptic pair. Therefore, the [ G, π ( S,θ ) ] G -type constructed by thedatum (cid:0) ( G i ) di =0 , y, ( r i ) di =0 , ρ ′ , ( φ i ) di =0 (cid:1) is regular supercuspidal.We determine the structure of Hecke algebras associated with regular supercuspidal types. So,let ( S, θ ) be a regular tame elliptic pair and ( ◦ K, ◦ ρ ) be as above. We consider the Hecke algebraˇ H ( G, ◦ ρ ) associated with the type ( ◦ K, ◦ ρ ). Proposition 5.4.
The support of ˇ H ( G, ◦ ρ ) is S ( F ) ◦ K .Proof. By Proposition 4.1, it is enough to show that g ∈ G ( F ) [ y ] intertwines ◦ ρ − if and onlyif g ∈ S ( F ) G ( F ) y . If g ∈ S ( F ) G ( F ) y , it is obvious that g intertwines ◦ ρ − . Conversely,suppose that g ∈ G ( F ) [ y ] intertwines ◦ ρ − . By the construction of ◦ ρ − , there exists g ∈ G ( F ) y so that gg intertwines ◦ κ ( S,φ − ) . Then, as in the proof of Proposition 5.2, we conclude that gg ∈ S ( F ) G ( F ) y, , and so g ∈ S ( F ) G ( F ) y . (cid:3) We define S ( F ) b = ◦ K ∩ S ( F ) = G ( F ) y ∩ S ( F ), which is the unique maximal compact subgroupof S ( F ). Corollary 5.5.
The algebra ˇ H ( G, ◦ ρ ) is isomorphic to the group algebra C [ S ( F ) /S ( F ) b ] of S ( F ) /S ( F ) b .Proof. Since ◦ ρ − is the restriction of ind S ( F ) G ( F ) y S ( F ) G ( F ) y, κ ( S,φ − ) to G ( F ) y , we can extend ◦ ρ − to S ( F ) G ( F ) y . Therefore, we can extend ◦ ρ to S ( F ) ◦ K . For [ g ] = g ◦ K ∈ S ( F ) ◦ K/ ◦ K , we define f [ g ] ∈ ˇ H ( G, ◦ ρ ) by f [ g ] ( x ) = ( ◦ ρ ( x ) ( x ∈ [ g ])0 (otherwise) . Then, as a vector space ˇ H ( G ( F ) , ◦ ρ ) = M [ g ] ∈ S ( F ) ◦ K/ ◦ K C f [ g ] and for [ g ] , [ h ] ∈ S ( F ) ◦ K/ ◦ K , f [ g ] ∗ f [ h ] = f [ gh ] . Hence ˇ H ( G ( F ) , ◦ ρ ) is isomorphic to the groupalgebra of S ( F ) ◦ K/ ◦ K ≃ S ( F ) /S ( F ) b . (cid:3) References [1] Jeffrey D. Adler and Manish Mishra,
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