On the construction of tame supercuspidal representations

Let F be a non-archimedean local field of odd residual characteristic. Let G be a (connected) reductive group over F that splits over a tamely ramified field extension of F. We revisit Yu's construction of smooth complex representations of G(F) from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in [Yu01], whose proofs relied on a typo in a reference.


Introduction
In 2001, Yu ([Yu01]) proposed a construction of smooth complex supercuspidal representations of p-adic groups that since then has been widely used, e.g. to study the Howe correspondence, to understand distinction of representations of p-adic groups, to obtain character formulas and to construct an explicit local Langlands correspondence. However, Loren Spice noticed recently that Yu's proof relies on a misprinted 1 (and therefore false) statement in [Gér77] and therefore it became uncertain whether the representations constructed by Yu are irreducible and supercuspidal. In the present paper we illustrate the significance of this false statement on Yu's proof by providing a counterexample to Proposition 14.1 and Theorem 14.2 of [Yu01]. Proposition 14.1 and Theorem 14.2 are the main intertwining results in [Yu01] that form the heart of the proof. We also offer a different argument to show that nevertheless Yu's construction yields irreducible supercuspidal representations.
Let F be a non-archimedean local field of residual characteristic p = 2. Let G be a (connected) reductive group that splits over a tamely ramified field extension of F . In this paper, we first describe the construction of Yu's representations in a way that we find more convenient for our purpose and then provide a proof that these representations are supercuspidal. All representations arise via compact induction from an irreducible representation ρ of a compact-mod-center open subgroup K of G(F ). Our proof only relies on the first part of Yu's proof and provides a shorter, alternative second part that does not rely on [Yu01, Proposition 14.1 and Theorem 14.2] and the misprinted version of [Gér77,Theorem 2.4(b)]. Yu's approach consists of following a strategy already employed by Bushnell-Kutzko that required to show that a certain space of intertwining operators has dimension precisely one, i.e., in particular, is non-trivial. Our approach does not require such a result. Instead we use the structure of the constructed representation including the structure of Weil-Heisenberg representations, and the Bruhat-Tits building to show more directly that every element that intertwines ρ is contained in K, which implies the desired result. Our proof relies also less heavily on tameness assumptions, and our aim is to use a modification of it for the construction of supercuspidal representations beyond the tame setting when Yu's construction is not exhaustive.
Note that Yu's construction yields all supercuspidal representations if p does not divide the order of the Weyl group of G ( [Fin21,Kim07]), a condition that guarantees that all tori of G split over a tamely ramified field extension of F .
In the last section we provide a counterexample to [Yu01,Proposition 14.1 and Theorem 14.2] by considering the group G = Sp 10 together with a twisted Levi subgroup G ′ of shape U(1) × Sp 8 and a well chosen point in the Bruhat-Tits building of G ′ .
Conventions and notation. Let F be a non-archimedean local field of residual characteristic p = 2. We denote by O the ring of integers of F , and by P the maximal ideal of 1 As Loren Spice pointed out, the statement of [Gér77, Theorem 2.4.(b)] contains a typo. From the proof provided by [Gér77] one can deduce that the stated representation of P (E + , j)H(E ⊥ + , j) (i.e. the pull-back to P (E + , j)H(E ⊥ + , j) of a representation of SH(E 0 , j 0 ) as in part (a')) should be tensored with χ E+ ⋉ 1 before inducing it to P (E + , j)H(E, j) in order to define π + (using the notation of [Gér77]).
O. The residue field O/P is denoted by F q , where q denotes the number of elements in F q . When considering field extensions of F in this paper, we mean algebraic field extensions of F and view them as contained in a fixed algebraic closure F of F . If E is a field extension of F , then we write E ur for the maximal unramified extension of E and E sep for the separable closure of E.
All reductive groups in this paper are required to be connected.
For a reductive group G defined over F we denote by B(G, F ) the (enlarged) Bruhat-Tits building ( [BT72,BT84]) of G over F , by Z(G) the center of G and by G der the derived subgroup of G. If T is a maximal, maximally split torus of G E := G × F E for some field extension E over F , then A (T, E) denotes the apartment of T inside the Bruhat-Tits building B(G E , E) of G E over E. Moreover, we write Φ(G E , T ) for the roots of G E × E F with respect to T F . We let R = R ∪ {r+ | r ∈ R} with its usual order, i.e. for r and s in R with r < s, we have r < r+ < s < s+. For r ∈ R ≥0 , we write G x,r for the Moy-Prasad filtration subgroup of G(F ) of depth r at a point x ∈ B(G, F ). For r ∈ R, we write g x,r for the Moy-Prasad filtration submodule of g = Lie G(F ) of depth r at x, and g * x,r for the Moy-Prasad filtration submodule of depth r at x of the linear dual g * of g. If x ∈ B(G, F ), then we denote by [x] its image in the reduced Bruhat-Tits building and we write G [x] for the stabilizer of [x] in G(F ).
We call a subgroup G ′ of G (defined over F ) a twisted Levi subgroup of G if (G ′ ) E is a Levi subgroup of G E for some (finite) field extension E of F . If G ′ splits over a tamely ramified field extension of F , then, using (tame) Galois descent, we obtain an embedding of the corresponding Bruhat-Tits buildings B(G ′ , F ) ֒→ B(G, F ). This embedding is only unique up to some translation, but its image is unique, and we will identify B(G ′ , F ) with its image in B(G, F ). All constructions in this paper are independent of the choice of such an identification.
Let H be a group and χ a character of H. Then we denote by C χ the one dimensional complex representation space on which H acts via χ. We also write 1 to denote the one dimensional trivial complex representation. If K is a subgroup of H, h ∈ H, and ρ a representation of K, then we write h K to denote hKh −1 and define h ρ( Throughout the paper we fix an additive character ϕ : F → C * of F of conductor P and a reductive group G that is defined over our non-archimedean local field F and that splits over a tamely ramified field extension of F . All representations of G(F ) in this paper have complex coefficients and are required to be smooth.
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Construction of representationsà la Yu
In this section we recall Yu's construction of representations but formulated in a way that is better adapted to our proof of supercuspidality.

The input
The input for Yu's construction of supercuspidal representations of G(F ) (using the conventions from [Fin21], see Remark 2.4 for a comparison of Yu's notation with ours) is a tuple ( . . G n+1 are twisted Levi subgroups of G that split over a tamely ramified extension of F (e) φ i , for 1 ≤ i ≤ n, is a character of G i+1 (F ) of depth r i that is trivial on (G i+1 ) x,r i + satisfying the following conditions Remark 2.2. Note that for each apartment A of B(G n+1 , F ), there exists a maximal torus T of G n+1 that splits over a tamely ramified extension E of F such that A ⊂ A (T, E) (see, e.g., [Yu01, §2, page 585-586], which is based on [BT84] and [Rou77]). In particular, there exists a maximal torus T of G n+1 that splits over a tamely ramified extension E of F such that x ∈ A (T, E).
Remark 2.3. By (the proof of) [MP96, Proposition 6.8] requiring that the image of the point x in B(G der n+1 , F ) is a vertex and that ρ| (G n+1 ) x,0 is a cuspidal representation of (G n+1 ) x,0 /(G n+1 ) x,0+ is equivalent to requiring that c-ind ρ is an irreducible supercuspidal representation. When n = 0, then the tuple ((G i ) 1≤i≤n+1 , x, (r i ) 1≤i≤n , ρ, (φ i ) 1≤i≤n ) consists only of the group G = G 1 = G n+1 , a point x ∈ B(G, F ) whose image in B(G der , F ) is a vertex and an irreducible representation ρ of G [x] that is trivial on G x,0+ and such that its restriction ρ| G x,0 is a cuspidal representation of G x,0 /G x,0+ . This case recovers the depth-zero supercuspidal representations.
Remark 2.4. We use the conventions for notation from [Fin21] instead of from [Yu01]. The notation in [Yu01] (left hand side) can be recovered from ours (right hand side) as follows: where r π = r 1 if n ≥ 1 and r π = 0 if n = 0. Yu's convention has the advantage that it is adapted to associating a whole sequence of supercuspidal representations to a given datum (by only considering the groups G i , G i+1 , . . . , G n+1 ), while our convention is more natural when recovering the input from a given representation as can be seen in [Fin21]. We have chosen our convention for this paper as it has the advantage that our induction steps below start with G 1 and move from G i to G i+1 . Moreover, using our notation we do not have to impose a condition on φ d depending on whether r d−1 < r d or r d−1 = r d in Yu's notation, see [Yu01, page 590 D5]. Hence the input looks more uniform. (Note that our condition G i = G i+1 in (iv) could be removed by extending the notion of G i -generic to the case G i = G i+1 .)

The construction
The (smooth complex) representation π of G(F ) that Yu constructs from the given input ( In order to define K and ρ we introduce the following notation. For r ≥ r ′ ≥ r 2 > 0 ( r, r ′ ∈ R) and 1 ≤ i ≤ n, we choose a maximal torus T of G i+1 that splits over a tamely ramified extension E of F and such that x ∈ A (T, E). Then we define where U α (E) x,r denotes the Moy-Prasad filtration subgroup of depth r (at x) of the root group U α (E) ⊂ G(E) corresponding to the root α. We define (g i ) x, r, r ′ analogously for [Yu01], and Yu ([Yu01, p. 585 and p. 586]) shows that this definition is independent of the choice of T and E.
We set In order to define κ we need some additional notation.
for some maximal torus T of G i+1 that splits over a tame extension E of F with x ∈ A (T, E), and the surjection g i+1 ⊕ r ′′ ։ g i+1 sends r ′′ to zero. (Recall that g = Lie(G)(F ), and (g E ) α denotes the E-subspace of Lie(G)(E) on which the torus acts via α.) and equip it with the pairing ·, · i defined by a, b i =φ i (aba −1 b −1 ). Then Yu shows in ([Yu01, Proposition 11.4]) that there is a canonical special isomorphism Then we define the space V κ underlying the representation κ to be n i=1 V ω i . If n = 0, then the empty tensor product should be taken to be a one dimensional complex vector space and κ is the trivial representation. In order to describe the action of K on each V ω i for n ≥ 1, we describe the action of (G i ) x,r i , r i 2 for 1 ≤ i ≤ n and of (G n+1 ) [x] separately.
with pairing a, b i =φ i (aba −1 b −1 ) (after choosing a from now on fixed isomorphism between the p-th roots of unity in C * and F p ) and composing this map with the Weil representation (defined in [Gér77]). Here the map from , which (together with the special isomorphism j i ) yields a symplectic action in the sense of [Yu01, §10] by [Yu01, Proposition 11.4].
Then the resulting actions of (G i ) x,r i , r i 2 for 1 ≤ i ≤ n and (G n+1 ) [x] agree on the intersections and hence yield a representation κ of K on the space V κ .
The representation π = c-ind G(F ) K ρ⊗κ is the smooth representation of G(F ) that Yu attaches to the tuple ((G i ) 1≤i≤n+1 , x, (r i ) 1≤i≤n , ρ, (φ i ) 1≤i≤n ), and we prove in the next section that π is an irreducible, supercuspidal representation.
3 Proof that the representations are supercuspidal We keep the notation from the previous section to prove the following theorem in this section.  Here we use an alternative and shorter approach to prove Theorem 3.1 that uses ideas from the first part of Yu's paper ([Yu01, Theorem 9.4]), but that avoids the second part that relies on the misprinted version of the theorem in [Gér77]. In particular, we do not use [Yu01, Proposition 14.1 and Theorem 14.2].
In order to show that c-ind G(F ) K ρ is irreducible, we first observe that ρ is irreducible.
We first prove by induction on i that ⊗ i j=1 V ω j is an irreducible representation of K i via the action described in Section 2.5. For i = 0, we take ⊗ i j=1 V ω i to be the trivial one dimensional representation and the statement holds. Now assume the induction hypothesis that However, since Heisenberg representations are irreducible, V ω i is irreducible as a representation of (G i ) x,r i , r i 2 ⊂ K i , and therefore V ′′ = V ω i . Thus ⊗ i j=1 V ω j is an irreducible representation of K i , and by induction the representation κ is an irreducible representation of K n .
Since K n acts trivially on ρ, every irreducible K-subrepresentation of ρ = ρ ⊗ κ has to be of the form ρ ′ ⊗ κ for an irreducible subrepresentation ρ ′ of ρ. As ρ is irreducible when restricted to (G n+1 ) [x] ⊂ K, we deduce that ρ is an irreducible representation of K.
The remaining proof of Theorem 3.1 is concerned with showing that if g intertwines ρ, then g ∈ K, which then implies that ind G(F ) K ρ is irreducible and hence supercuspidal. Our proof consists of two parts. The first part is concerned with reducing the problem to considering g ∈ G n+1 (F ) using the characters φ i and is essentially [Yu01, Corollary 4.5]. The second part consists of deducing from there the theorem using the depth-zero representation ρ together with the action of suitably chosen subgroups of higher depth and employing knowledge about the structure of Weil-Heisenberg representations. This is where our approach deviates crucially from Yu's approach by avoiding the wrong [Yu01, Proposition 14.1 and Theorem 14.2] in favor of a much shorter argument. For the first part, we will use the following result of Yu ([Yu01, Theorem 9.4]).

Proof of Theorem 3.1.
Recall that ρ is irreducible by Lemma 3.3. Thus, in order to show that c-ind G(F ) K ρ is irreducible, hence supercuspidal, we have to show that if g ∈ G(F ) such that then g ∈ K, where g K denotes g Kg −1 and g ρ(x) = ρ(g −1 xg).
Fix such a g ∈ G(F ) satisfying Hom K∩ g K ( g ρ, ρ) = {0}, and define We first prove by induction that g ∈ K n G n+1 (F ) K n using Lemma 3.4 (which is (part of) [Yu01, Theorem 9.4]). This is essentially [Yu01, Corollary 4.5], but we include a short proof for the convenience of the reader. Let 1 ≤ i ≤ n and assume the induction hypothesis that g ∈ K i−1 G i (F ) K i−1 , which is obviously satisfied for i = 1. We need to show that g ∈ K i G i+1 (F ) K i . Since K i−1 ⊂ K i ⊂ K, we may assume without loss of generality that g ∈ G i (F ). Recall that by construction ρ| (G i ) x,r i , r i 2 + = Id and κ| (G i ) x,r i , r i 2 + = n j=1φ j · Id. Thus by restriction of the action in Equation (1) to (G i ) x,r i , r i 2 + ∩ g (G i ) x,r i , r i 2 + we conclude that g intertwines ( n j=1φ j )| (G i ) x,r i , r i 2 + . By the definition ofφ j in Section 2.5, we have that (2) Therefore we obtain that g also intertwinesφ i | (G i ) x,r i , r i 2 + . By Lemma 3.4 (which is (part of) [Yu01, Theorem 9.4]) we conclude that g ∈ This finishes the induction step and therefore we have shown that g ∈ K n G n+1 (F ) K n .
In the remainder of the proof we exhibit a subspace V ′ κ ⊂ V κ such that V f ⊂ V ρ ⊗ V ′ κ and prove that the action of U on V ′ κ via κ is given by φ · Id. Hence, since U also acts via φ · Id on V f ⊂ V ρ ⊗ V ′ κ , we deduce that (ρ|Ū , V ρ ) contains the trivial representation, which contradicts that ρ| (G n+1 ) x,0 is cuspidal (see Condition (iii) of the input in Section 2.1).
Let T be a maximal torus of G that splits over a tamely ramified extension E of F such that x and g.x are contained in A (T, E). (Such a torus exists by Remark 2.2.) Let λ ∈ X * (T ) ⊗ Z R = Hom F (G m , T F ) ⊗ Z R such that g.x = x + λ, and observe thatŪ is the image of Since U i acts trivially via ρ on the space V ρ underlying the representation of ρ and U i acts via 1≤j≤n κ ⊂ V κ that we are looking for. In order to study the subspace V U i ω i for 1 ≤ i ≤ n, we recall that we write and equip V i with the pairing ·, · i defined by a, b i =φ i (aba −1 b −1 ) (using the above fixed identification of the p-th roots of unity in C * with F p ). We define the space V + i to be the image of i is a non-degenerate subspace of V i . We denote by P i ⊂ Sp(V i ) the (maximal) parabolic subgroup of Sp(V i ) that preserves the subspace V + i and that therefore also preserves V + i ⊕ V 0 i . We obtain a surjection pr i,0 : i is the Heisenberg group with underlying set V i × F p that is attached to the symplectic F p -vector space V i with pairing ·, · i , and note that the subset V 0 i × F p ⊂ V i × F p forms a subgroup, which is the Heisenberg group (V 0 i ) ♯ attached to the symplectic vector space V 0 i with the (restriction of the) pairing ·, · i . We denote by V 0 ω i a Weil-Heisenberg representation of Sp(V 0 i ) ⋉ (V 0 i ) ♯ corresponding to the same central character as the central character of V ♯ i acting on V ω i (which in turn corresponds to the characterφ i | (G i ) x,r i , r i 2 + via the special isomorphism j i ). By [Gér77, Theorem 2.4.(b)] the restriction of the Weil-Heisenberg is a one dimensional space on which the action of P i is given by a quadratic character 2 χ V + i that factors through the projection pr i,+ : Then by Yu's construction of the special isomorphism j i : as a representation of P i .
Note that the image ofŪ Sp(V i ) in GL(V + i ) under the projection pr i,+ : P i → GL(V + i ) is unipotent sinceŪ is unipotent. Hence pr i,+ (Ū Sp(V i ) ) is contained in the commutator subgroup of GL(V + i ), and χ V + i | pr i,+ (Ū Sp(V i ) ) is trivial. Moreover, we observed above that pr i,0 (Ū Sp(V i ) ) = . Recall that the action of U on (V ω i ) U i is given by the product of φ i | U with the above Weil representation construction, see Section 2.5, page 7. Hence U acts on (V ω i ) U i via the character φ i | U . Since we proved above that U acts via we deduce that there exists a non-trivial subspace V ρ,f of V ρ on which U acts trivially. Hence ρ|Ū contains the trivial representation, which contradicts that ρ| (G n+1 ) x,0 is cuspidal. of the section more precisely, let G ′ be a tamely ramified twisted Levi subgroup of G, let x ∈ B(G ′ , F ), and φ a character of G ′ (F ) that is G-generic relative to x of depth r for some r ∈ R >0 , i.e. we are in the setting of [Yu01,§ 14]. Following [Yu01], we set

A counterexample
whereφ is defined as in [Yu01,§4 and §9], see also page 6 of this paper, and we denote by φ the representation of K ⋉J which is the pull back of the Weil representation of Sp(J/J + )⋉(J/N) via the symplectic action given by [Yu01,Proposition 11.4], see also page 7 of this paper.
In this section, we provide an example for G ′ ⊂ G, x and φ as above and g ∈ G ′ (F ) such that dim Hom (K∩ g K)⋉(J∩ g J) ( g φ, φ) = 0.
Following [Yu01,§14] we denote by φ ′ the representation of KJ whose inflation inf φ ′ to K ⋉ J yields inf(φ| K ) ⊗ φ. By the discussion in [Yu01] immediately following Theorem 14.2 (see also Corollary 4.3 below), Equation (6)  Consider the case G = Sp 10 over F corresponding to the symplectic pairing given by 0 J 5 −J 5 0 where We assume that the residue field of F is F p for some prime number p > 10.
We set G ′ to be the centralizer Cent G (X) of X in G. Note that is a twisted Levi subgroup of G = Sp 10 (with anisotropic center).
Hence val(X(H α )) = val(±2̟ −1 ) = − 1 2 for all α ∈ Φ(G, T ′ ) \ Φ(G ′ , T ′ ), where val denotes the valuation of F with image Z ∪ {∞}. Moreover, X is contained in (g ′ ) * y,− 1 2 for any point y ∈ B(G ′ , F ) and its restriction to the Lie algebra of the identity component Z(G ′ ) • of the center of G ′ is contained in Lie * (Z(G ′ ) • ) − 1 2 . Since p > 10, i.e. p does not divide the order of the Weyl group of Sp 10 , we conclude that X is G-generic of depth 1 2 . Let E be a finite, tamely ramified extension of F that contains √ 2̟. Then x is a vertex in B(G, E) and it is an easy calculation to check that for the point y = (0, 0, 0, 1 4 , 1 4 ) ∈ A (T, F ) ⊂ A (T, E).
The element X yields a linear map from g ′ x, 1 2 to O that sends g ′ x, 1 2 + to ̟O and defines a character of G ′ x, 1 2 that is trivial on G ′ . Since U(1) is abelian, we can extend this character to a character of G ′ (F ) (trivial on Sp 8 (F ) ⊂ G ′ (F )), which we denote by φ. Since X is G-generic of depth r = 1 2 (for the pair G ′ ⊂ G), the character φ is G-generic relative to x of depth r in the sense of Yu ([Yu01, § 9]). Then dim Hom (K∩ g K)⋉(J∩ g J) ( g φ, φ) = 0.

Proof.
Using the standard coordinates we define the groups Note that H 23 ≃ GL 2 (O) ≃ H 45 , and gH 23 g −1 = H 45 and gH 45 g −1 = H 23 , hence H 23 ∈ K ∩ g K. Moreover, the image of H 23 and the image of H 45 in G ′ x,0 /G ′ x,0+ are both isomorphic to GL 2 (F p ).