Types and unitary representations of reductive p-adic groups

We prove that for every Bushnell–Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. Moreover, we show that every irreducible smooth G-representation contains a rigid type. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.

Let e ∈ H be an idempotent. Examples of idempotents include e K = μ(K ) −1 δ K , where δ K is the indicator function for a compact open subgroup K in G. Then eHe is a associative subalgebra of H and eHe has e as its identity element. Let IrrG denote the set of isomorphism classes of irreducible smooth complex G-representations and let Irr e G be the subset of irreducible representations (π, V ) such that eV = π(e)V = 0. Let Irr(eHe) denote the set of (isomorphism classes of) simple eHe-modules. We have a canonical bijection Irr e G → Irr(eHe), V → V e . (1.1) The operation * is a conjugate-linear anti-involution of H. Suppose that e is self-adjoint, i.e., e * = e. The algebra eHe inherits the operation * and has a natural structure of a normalized Hilbert algebra. The results of Bushnell et al. [14] give an identification between the supports of the Plancherel measures under the bijection (1.1). More precisely, let G denote the unitary dual of G (the topological space of irreducible unitary representations of G on Hilbert spaces) with the Plancherel measureμ dual to μ. Let G r denote the support of μ, the space of tempered irreducible G-representations. Denote by G r (e) the set of representations (π, V ) ∈ G r such that π(e) = 0. On the other hand, eHe has a C * -algebra completion C * r (eHe) whose dual C * r (eHe) carries a Plancherel measureμ eHe . By [14, Theorem A], the bijection (1.1) induces a homeomorphism m e : G r (e) → C * r (eHe), (1.2) such that for every Borel set S of G r (e), μ(S) = e(1) μ eHe ( m e (S)). In other words, (1.1) induces a natural bijection between irreducible tempered representations.
It is natural to ask if in fact (1.1) induces a bijection of irreducible unitary representations. (We are identifying here preunitary smooth G-representations with unitary G-representations.) In complete generality, this is clearly false, as seen, for example, by taking e = e K 0 where K 0 is a maximal special compact open subgroup. In that case, if V is any irreducible representation with a *invariant Hermitian form such that V K 0 = 0, then V K 0 is automatically a one-dimensional unitary e K 0 He K 0 -module.
For an idempotent e, define C e (G) to be the full subcategory of C(G) consisting of representations (π, V ) such that HeV = V . Following [12], we say that e is special if C e (G) is closed relative to the formation of G-subquotients. This is equivalent [12, 3.12] to the requirement that the functor m e : C e (G) → eHe − Mod, V → eV (1.3) is an equivalence of categories.
Let K be a compact open subgroup and (ρ, W ) a smooth irreducible (finite dimensional) K-representation. Let (ρ ∨ , W ∨ ) be the contragredient representation. The pair (K, ρ) is called a type if e ρ (x) = μ(K) −1 dim W tr W (ρ(x −1 ))δ K (x), x ∈ G, (1.4) is a special idempotent. Our main result is the following Theorem 1.1 Suppose that e ∈ H is a self-adjoint special idempotent such that e = e ρ for a type (K, ρ). If (K, ρ) is rigid (in the sense of Definition 3.9), then m e induces a bijection between unitary representations in C e (G) and unitary eHe-modules.
In addition we show

Theorem 1.2 Every irreducible smooth G-representation contains a rigid type.
The rigid type in Theorem 1.2 can be taken to be either a level zero G-type as in [34], or a positive level unrefined minimal K -type, in the sense of [35].
In the particular situation when G is G L(n, F) or one of its inner forms, it is immediate to see that every type is rigid. This has to do with the fact that there is only one conjugacy class of maximal compact open subgroups.
Define the ρ-spherical Hecke algebra H(G, ρ) to be the convolution algebra of locally constant compactly supported functions f : G → End C (W ∨ ) such that f (k 1 xk 2 ) = ρ ∨ (k 1 ) f (x)ρ ∨ (k 2 ), k 1 , k 2 ∈ K, x ∈ G. This algebra is a normalized Hilbert algebra and it is Morita equivalent with e ρ He ρ . Set V ρ = Hom K [ρ, V ]; this is an H(G, ρ)-module. As a consequence, we obtain the following equivalence. These results represent a generalization of the Barbasch and Moy preservation of unitarity [4,5] which is the case e = e I where I is an Iwahori subgroup for a split group G. The ideas of [4,5] were extended further in [3] to situations where H(G, ρ) is known to be isomorphic as a * -algebra to an affine Hecke algebra with geometric parameters in the sense of Lusztig [31,32]. The [4,5] proof in the Iwahori case (and in the generalization [3]) is based on three main ingredients: (a) Vogan's signature character [43]; (b) The fact that m e maps irreducible tempered representations to irreducible tempered representations; (c) A certain linear independence result proved using Kazhdan-Lusztig theory [29] and a technical reduction to Lusztig graded affine Hecke algebra [5,31].
In particular, this approach is dependent on the knowledge that e I He I is a specialization of the generic affine Hecke algebra (possibly with parameters).
In the present paper, the proof of Theorem 1.1 still relies on Vogan's signature character, but instead of considering K -signature characters with respect to the maximal special compact open subgroup K = K 0 , we consider the signature characters with respect to all conjugacy classes of (maximal) compact open subgroups. The second and essential difference is that the necessary linear independence is obtained as a consequence of the trace Paley-Wiener Theorem proved in [8], see also the work of Henniart and Lemaire [23] and [19] for more recent accounts and generalizations. We also need to make use of the interplay between the rigid cocenter and the rigid representation space, in the sense of [18,19]. We emphasize that for this argument, we do not need to know the precise structure of the algebra e ρ He ρ or the statement (b) above, nor do we need the reduction to real infinitesimal character for unitary representations of affine Hecke algebras from [5].
We give a brief outline of the paper. In Sect. 2, we present the definitions and main properties of the K -signature character of a Hermitian G-representation and adapt Vogan's signature theorem to our setting. The main result of the section is Theorem 2.7. In Sect. 3, we explain the ideas around the cocenter and trace Paley-Wiener theorem that we need for our applications. The main points are the rigid trace Paley-Wiener theorems for Bernstein components, Corollaries 3.7 and 3.8. In Sect. 4, we use these results to give proofs of Theorem 1.1 and of Corollary 1.3. Finally, in Sect. 5, we discuss the notion of rigid type in more detail and prove Theorem 1.2.

Vogan's signature character
In this section, we recall the definition of the multiplicity K -character and signature K -character for an admissible G-representation (π, V ) , where K is a compact open subgroup, and explain an analogue of a formula of Vogan for the signature character [2,43].

The signed Grothendieck group
We begin by recalling the idea of the Grothendieck group of modules with nondegenerate Hermitian forms from [2]. As in the introduction, H denotes the Hecke algebra of a reductive p-adic group G. A star operation κ : H → H is a conjugate linear involutive algebra anti-automorphism. The main example for us in this paper is We say that V is Hermitian if it has a nondegenerate κ-invariant Hermitian form.
Suppose K is a compact open subgroup of G. Every smooth irreducible K -representation (μ, E μ ) is finite dimensional, hence unitary, and we fix a positive definite K -invariant form , μ on E μ . Let K denote the set of equivalence classes of such K -representations. If V is admissible, denote by V (μ) = π(e μ )V the isotypic component of μ in V , and by If V has an invariant Hermitian form, let , V (μ) denote the restriction of the form , V to V (μ). This induces a Hermitian form , to be the dimension of the positive definite subspace of (V μ , , μ V ), the dimension of the negative definite subspace, and the dimension of the radical, respectively: If V is admissible, let V h,κ denote the Hermitian dual module. This is an admissible module and V h,κ If U is an H-submodule of (V, , V ), denote by U ⊥ the orthogonal complement of U in V with respect to , V .
If R is any admissible module, define the hyperbolic form on R ⊕ R h,κ , i.e., the nondegenerate κ-Hermitian form , hyp : For every compact open subgroup K , we have a well defined signature homomorphism and a multiplicity homomorphism Here R(H) denotes the ordinary Grothendieck group of admissible Hmodules, and Fun[ ] denotes the set of functions.
As in [2], a better way to define the signature homomorphism is by introducing the signature Grothendieck ring W of Hermitian finite dimensional C-vector spaces. This is the Hermitian Grothendieck group of finite dimensional C-vector spaces together with the tensor product.
The following result expresses the Hermitian form of an admissible module in terms of the Hermitian forms of its composition factors.
In particular, for every compact open subgroup K , the K -signature character of V is In this notation, a nondegenerate Hermitian form , on V is positive definite fi the coefficient of s in the signature is zero.

Langlands data
Fix a minimal parabolic subgroup P 0 = M 0 N 0 defined over F. Let A 0 ⊂ M 0 be a maximal split torus over F. We will denote by P 0 the F-points of For a standard pair (P, A), we regard a * as a real subspace of a * 0 and similarly a ⊂ a 0 . Let H : M → a be the Harish-Chandra map [16,§XI.1.13]. Choose an inner product (, ) on a * 0 which is invariant under the action of the Weyl group of A 0 in G. Let (P, A) denote the root system of (P, A). Define a * ,+ = {ν ∈ a | (ν, α) > 0 for all α ∈ (P, A)}.
Let {α i } be the set of simple roots in (P, A) and let {ω i } be a basis ( , )-dual to {α i } in the span of {α i }. Define a partial order ≤ on a * , by setting ν ≤ λ whenever (λ − ν, ω i ) ≥ 0 for all i.
For a standard parabolic subgroup P = M N of G, denote by i G P the functor of normalized parabolic induction and by r G P the normalized Jacquet functor (the normalized functor of N -coinvariants). Let M 0 be the subgroup generated by all compact subgroups of M, an open normal subgroup of M. Let δ P be the modulus function. Denote by R(G) the Grothendieck group of admissible Grepresentations. We may regard i G P as a map i G P : R(M) → R(G), independent of P ⊃ M, and therefore it may be denoted by i G M in this situation. If (σ, U σ ) is an admissible M-representation, and ν ∈ a * c , define the induced representation: with the G-action given by right translations.
A Langlands datum is a triple (P, σ, ν), where (P, A) is a standard pair, P = M N , σ is an irreducible tempered representation of M 0 , and ν ∈ a * ,+ . The following theorem is the Langlands classification in the p-adic case, and summarizes [

Theorem 2.3 Let (P, σ, ν) be a Langlands datum andP = MN the opposite parabolic to P.
(1) The integral operator (2.5) converges absolutely and uniformly in g on compacta, and it is an intertwining operator with respect to the G-actions.

An invariant Hermitian form
which induces an isomorphism, still denoted a σ,w between I (P, σ, −ν) and I (P, σ, ν). We emphasize that a σ,w does not depend on ν. Let , σ be a positive definite * M -invariant Hermitian form on U σ . Define a possible degenerate *invariant Hermitian form on I (P, σ, ν) by The radical of , P,σ,ν,w is ker j (ν). Taking the quotient by the radical induces an nondegenerate * -invariant Hermitian form on J (P, σ, ν). Since J (P, σ, ν) is irreducible, we know that this form is unique up to scalar multiple.

The Bernstein center
We recall the definitions of the Bernstein center and the inertial equivalence class. Let M be a Levi subgroup of G with the subgroup M 0 as before. It is known that M/M 0 is a free abelian group of finite rank. A character χ : Consider the set of pairs (M, σ ), where M is a Levi subgroup and σ is a supercuspidal representation of M. Define the relation of equivalence such pairs: Define the weaker relation of inertial equivalence on the same pairs: defines a connected component of (G) which will be denoted (G) s . It has the structure of a complex affine algebraic variety as the quotient of X (M) by a finite group W s . The decomposition makes (G) into a complex algebraic variety with infinitely many connected components.
As it is well known, if (π, V ) ∈ IrrG, there exists a unique up to equivalence cuspidal pair (L , σ ) such that π is a subquotient of i G P (σ ). Then (L , σ ) defines a point θ ∈ (G), which is called the infinitesimal character of π . The infinitesimal character map inf-char : IrrG → (G) (2.9) is onto and finite to one. The Bernstein center Z(G) is the algebra of regular functions on (G).
of functions supported on a finite number of components. Then each infinitesimal character θ ∈ (G) can be identified with an algebra homomorphism θ : Z(G) → C which is nontrivial on Z 0 (G). With this identification, on each irreducible smooth representation (π, V ), z ∈ Z(G) acts by (see [7, §2.13

Inertial support
We call the inertial class s = [L , σ ] the inertial support of π . Let C s (G) denote the full subcategory of C(G) consisting of representations π with the property that every subquotient of π has inertial support s. Then as abelian categories. If e ∈ H is a special idempotent, then the results of [12, 3.12] (see also [14, 3.6]) say that there exists a subset S(e) ⊂ B(G) such that Conversely, it is shown in loc. cit. that for every s ∈ B(G), there exists a special, self-adjoint idempotent e such that C e (G) = C s (G).

The K -character
Let K be a compact open subgroup. The goal is to express the K -signature of an irreducible smooth G-representation in terms of that of tempered modules. A first step is the simpler result for K -multiplicity functions.
and integers a i such that for every K , There are finitely many irreducible Grepresentations, all in C s (G), with infinitesimal character θ . Write them in terms of the Langlands classification as J (P, σ, ν) and fix an ordering on this set compatible with ≤ on the Langlands parameters ν. σ 1 ,0) , and I (P 1 , σ 1 , 0) is tempered and in C s (G). If I (P 1 , σ 1 , 0) is reducible then it decomposes into a direct sum of finitely many irreducible tempered representations, hence the claim is proved in the base case of the induction. Now let V = J (P, σ, ν) be arbitrary. If J (P, σ, ν) = I (P, σ, ν), the claim follows in the same way as before. Otherwise, let J i = J (P i , σ i , ν i ) be the composition factors of I (P, σ, ν) other than J (P, σ, ν). Then Since m K I (P,σ,ν) = m K I (P,σ,0) and λ J i < λ V , the proof is complete by induction.
Let (P, σ, ν) be a Langlands datum such that J (P, σ, ν) admits a nondegenerate * -invariant Hermitian form. Regard the standard module I (P, σ, ν) as a member of the continuous family I t = I (P, σ, tν), t ∈ R ≥0 . As it is well known, all of the representations in the family I t can be realized on the same space X . Explicitly, let K 0 be a maximal compact open subgroup such that the Iwasawa decomposition holds G = P K 0 . Then It is important to notice that for every compact subgroup K , the restriction of I t to K is independent of t. The Hermitian forms , P,σ,tν,w on I t can be regarded as an analytic family of Hermitian forms , t on X such that J (P, σ, tν) = I (P, σ, tν)/ ker , t , for t > 0.
The Jantzen filtration of I 1 (t = 1, obviously the same definition applies to an arbitrary point t 0 > 0) is: where X n consists of the vectors x ∈ X such that there exists a an analytic function satisfying the conditions: (a) f x takes values in a fixed finite dimensional subspace of X ; (c) for all y ∈ X , the function t → f x (t), y t vanishes at least to order n at t = 1.
Define a Hermitian form , n on X n by (2.14) The main result is the following theorem.

15)
where sig t denotes the K -signature of , t .
Before we get to the signature theorem, we need a lemma which will insure that the inductive algorithm terminates (see also [40,Lemma 2.12] for similar considerations). This means that Theorem 2.7 (The signature theorem) Let (π, V ) be an irreducible smooth G-representation with inertial support s. Suppose that V admits a *invariant Hermitian form. Then there exist irreducible tempered representations V 1 , . . . , V n in C s (G) and w 1 , . . . , w n ∈ W (see (2.2)) such that, for every compact open subgroup K , the K -signature of V is Proof The proof is the same as in loc. cit., except that we need to explain in our setting the independence of K , the Bernstein component s, and the fact that the algorithm terminates.
Write V in terms of the Langlands classification as V = J (P, σ, ν). As before, consider I t = I (P, σ, tν), t ∈ [0, 1]. Let t 1 < · · · < t r 1 be the points in (0, 1) where I t is reducible, and set t 0 = 0 and t r = 1. For 1 ≤ j ≤ r define For 1 ≤ j ≤ r , the induced representation I t j has the Jantzen filtration The subquotient X n t j / X n+1 t j has the nondegenerate form , j n with signature sig K ,n j . By Theorem 2.5, we have the recursion formulas for sig K j : (2.18) In particular, the signature of V is: As in [43, 3.38], this leads to (2.20) Notice first that in the right hand side of this formula, the signature of J t j does not appear for any j ≥ 1. The signature sig K 1 is the same as the signature of I (P, σ, 0), which is a possibly reducible tempered module, and its signature can be written uniquely in terms of the signatures of its irreducible tempered composition factors. For the rest of the terms, we want to proceed by induction on the length of the Langlands parameter ν = λ V .
One remarks, as in [43, Lemma 3.1] that if the conclusion of Theorem 2.7 holds for all the irreducible Hermitian composition factors of an admissible module Y which have a nondegenerate Hermitian form, then it holds for Y itself. This is because of Proposition 2.2 and Lemma 2.4. In particular, we may apply this observation to our situation, namely to the (possibly reducible) subquotients X n t j / X n+1 t j in the Jantzen filtrations, as long as by induction we may assume that the claim holds for the irreducible composition factors of I t j , other than J t j . Let V be an irreducible Hermitian composition factor of I (P, σ, t j ν) for some 1 ≤ j ≤ r , such that V = J (P, σ, t j ν). These are the irreducible representations whose signature contribute in the right hand side of (2.20). By Lemma 2.6, the Langlands parameter of V satisfies ||λ V || < ||tν j || − d s ≤ ||λ V || − d s < ||λ V ||. The induction hypothesis applies then and therefore the conclusion of the theorem holds for V as well, by (2.20). Notice that the algorithm terminates since every time the length of the Langlands parameter drops by at least d s > 0. Corollary 2.8 With the same notation as in Theorem 2.7:

Remark 2.9
The same formula in Theorem 2.7 holds with respect to different star operations κ as long as for every κ-Hermitian Langlands quotient J (P, σ, ν), we can define a natural (degenerate) κ-invariant form on I (P, σ, ν) whose radical is ker j (ν). In the setting of (g, K )-modules, an essential role is played by the signature theorem for the compact star operation [2]. We will explore a p-adic analogue in future work.

Rigid tempered representations
In this section, we explain several relevant results from [8,[19][20][21] which will allow us to sharpen the signature theorem 2.7. See also [9,37,42] for more details on the relation between the cocenter, K 0 (G), and characters of admissible representations. If A is an abelian group and R is a ring, we will denote A R = A ⊗ Z R.

The compact cocenter
Recall that R(G) denotes the Z-span of the set of isomorphism classes of irreducible smooth G-representations.

The compact representations space
Define R ind (G) Q ⊂ R(G) Q to be the Q-span of all induced modules i G P (σ ), σ ∈ R(L) Q , where P = G.
where P ranges over the set of parabolic subgroups, σ ∈ R(L), and χ ∈ X (L). Define the compact representations space (quotient): In the case when G is semisimple, this is the same notion as that of the rigid quotient from [19].

Lemma 3.2 The space R c (G) Q is spanned by the classes of irreducible tempered G-representations.
Proof By the Langlands classification, R(G) is spanned by the Langlands standard modules I (P, σ, ν), where (P, σ, ν) are Langlands date. It is clear that in R rigid (G) Q , we have I (P, σ, ν) ≡ I (P, σ, 0), the latter being a tempered representation of G.
The elliptic representation space is The space R ell (G) Q is also spanned by the images of the irreducible tempered modules (again by invoking the Langlands classification), and there is a natural surjection R rigid (G) Q → R ell (G) Q . Every irreducible discrete series representation gives a nonzero class in R ell (G) Q , see for example [28].

The trace map
Let K 0 (G) denote the Grothendieck group of finitely generated projective G-representations. Let denote the Hattori-Stallings trace map, see for example [21, §2.2].
The following result is essentially the abstract Selberg principle for p-adic groups proved in [10].
More is known about τ , for example, it is also proved in the [20] that (a) The map τ is injective, and (b) The space K 0 (G) Q is generated over Q by compactly induced modules where K ranges over the compact open subgroups of G and (μ, E μ ) over K .
We will not need these finer results.

Theorem 3.4 ([8, Theorem 1.2]) The trace map tr : H(G) → F(R(G)) given by f
In fact, Kazhdan's density theorem says that tr is also injective [27,28], but we will not use this result.

The rigid trace Paley-Wiener theorem
We now look at the restriction of tr to H c (G). If σ ∈ K , K a compact open subgroup, let χ σ : K → C denote its character. One may extend this by zero to an element of H(G) and regard χ σ as an element of H c (G). In fact, the compact cocenter H c (G) is spanned by χ σ as σ ranges over K for all compact open subgroups K . If π is an admissible G-representation, then it is well known that tr π( This means that we naturally have a map (3.5) The following result is proved in [19] in the more general setting of mod-l representations. It can also be deduced from [8] in combination with [20,21]. (1) tr c :

Bernstein components
We need to look at the action of the Bernstein center. Let U (G) denote the inverse limit algebra where for K ⊂ K , the inverse limit system is given by This is a unital associative algebra. As it is known [7], see also the exposition in [36], the algebra U (G) has the equivalent description as the convolution algebra of essentially compact distributions: (3.7) The Bernstein center can be identified as (3.8) The star operation * extends to U (G) by setting The Hecke algebra itself is a * -subalgebra (in fact, a two-sided ideal) of U (G).
The nontrivial step is the following:

Proposition 3.6 [21, Proposition 2.8]
For every idempotent element e ∈ Z(G), Here, we think of e ∈ End(H(G)) via e( f ) = e f , for f ∈ H(G). Proposition 3.6 says that for every f ∈ H c (G), Let R c (G, s) be the image of R(G, s) in R c (G). By the discussion above, we can also define H c (G, s) = H c (G)e s ⊂ H c (G).

Rigid pairs
Now suppose that (K, ρ) is a type with e ρ ∈ H(G) the corresponding idempotent. Let R(G, ρ) be the subspace of R(G) spanned by the irreducible objects in ( (G, ρ)) is surjective.
Proof The category C e ρ (G) is a direct product of finitely many Bernstein components [12, (3.12)]. Without loss of generality, we may assume that C e ρ (G) = for a finite set of Bernstein idempotents {e }. This implies that (3.12) in the obvious notation. Applying the Hattori-Stalling map τ and using Theorem 3.3: In Sect. 5, we will discuss important examples of rigid types.

The proofs of Theorem 1.1 and Corollary 1.3
We are now in position to prove Theorem 1.1.

A strong signature theorem
Firstly, we can sharpen Theorem 2.7 using the notion of rigid representations. Let B c (G, s) = {V 1 , . . . , V n } denote a basis of R c (G, s) Q consisting of irreducible tempered representations. From the previous section we know that R c (G, s) Q is finite dimensional.
Notice that, in this version, we cannot say that the scalars w i are integral, only rational.
Proof In light of Theorem 2.7, it is sufficient to show that the K -character of every irreducible tempered representation V with inertial support s can be written as a rational combination (independent of K ) of B c (G, s). But this is equivalent to writing V in R c (G, s) Q in terms of the basis B c (G, s).

Remark 4.2
An important question is if there is a natural basis B c (G, s) or more generally a basis of all of R c (G) or R c (G) Q . The analogy is that for (g, K )-modules of real reductive groups, a fundamental theorem of Vogan says that the set of irreducible tempered (g, K )-modules is a basis (over Z!) of the analogous quotient of the Grothendieck group of admissible (g, K )modules. In the setting of p-adic groups, we may decompose (see for example [20,Sect. 4] or [18,Proposition 6.5] for the Iwahori case) However, beyond this point, we do not know how to make canonical choices for the basis of R ell (M) Q,X (M) . In fact, the example of the two (Iwahori-spherical) tempered direct summands of the minimal principal series representation of SL(2, Q p ) induced from the unramified quadratic character appears to suggest that a canonical choice for irreducible elliptic representations may not be possible.
Since e is a self-adjoint idempotent an invariant Hermitian form on He ⊗ eHe U .

The proof of Theorem 1.1
The difficult part of the unitarity equivalence is to show that if (π, V ) carries a nondegenerate Hermitian * -invariant form , V such that , V,e is positive definite, then , V is positive definite, i.e., if π(e)V is unitary, then V is unitary. For this, we need to restrict to the case of types. Let (K, ρ) be a type as before and let (π, V ) be an irreducible representation in C e ρ (G).
For every compact open subgroup K ⊇ K, we have defined the notions of K -multiplicity and K -signature of V, , V . Define the notions of e ρ H (K )e ρmultiplicty and signature of π(e ρ )V, , V,e ρ : , (4.4) in exactly the same way. Using Theorem 4.1, write for certain a i , b i ∈ Z, independent of K . The signature sig e ρ H (K )e ρ π(e ρ )V is obtained by simply restricting to the K -types (μ, E μ ) such that π(e ρ )E μ = 0. Hence (4.7) The condition in (4.7) holds for all K ⊃ K, and therefore, when (K, ρ) is rigid, it can be rephrased in terms of the notation in Sect. 3 as saying that Then Corollary 3.8 implies that Now since {V i } is a basis of R c (G, ρ), we get that b i = 0 for all i, (4.10) and so , V is also positive definite.
Remark that the method of proof fails if one only considers the signature characters with respect to a fixed maximal compact open subgroup, for example the maximal special subgroup K 0 . The reason is that does not necessarily imply that n i=1 b i V i = 0 in R c (G, ρ). For example, take G = SL(2, Q p ), K 0 = SL(2, Z p ), and (K, ρ) = (I, 1 I ), where I ⊂ K 0 is an Iwahori subgroup. In that case, R c (G, e I ) is 3-dimensional, but there are only two irreducible K 0 -types with I -fixed vectors.

The proof of Corollary 1.3
Corollary 1.3 follows from Theorem 1.1 via a well-known argument, as applied for example in [14,Sect. 4] to deduce the preservation of Plancherel measures. We include it here for completeness. We follow the notation in [14]. Recall that a normalized Hilbert algebra A is an associative unital C-algebra with a star operation * and an inner product [ , ] A such that [1 A , 1 A ] = 1. We do not reproduce the axioms of compatibility between [ , ] A and * , but we refer to [14,Definition 3.1] for the details.
There are three normalized Hilbert algebras that enter in the picture for a type (K, ρ) such that e * ρ = e ρ . Recall that (ρ, W ) is an irreducible smooth K-representation with contragredient (ρ ∨ , W ∨ ). The first algebra is e ρ He ρ where the Hilbert product is The second is We fix a positive definite K-invariant form , ρ on W and define a star operation a → a * on E ρ via The inner product is [ , ] E ρ : (4.14) The same definitions apply to E ρ ∨ = End C [W ∨ ]. The transpose map a → a t defines an isomorphism of Hilbert algebras between E ρ and E ρ ∨ . Finally, H(G, ρ) is also a Hilbert algebra with involution h → h * defined by (4.15) and inner product . (4.16) As in [14, §4.4], define the tensor product (4.17) and endow it with a Hilbert algebra structure by using the product star operation and the product inner product . For every pair (h, a) ∈ H(G, ρ) × C E ρ , define the function

Rigid types
We retain the notation from Sect. 3.

Higher depth rigid types
Let I be an Iwahori subgroup and K a compact open subgroup.
is surjective and therefore is surjective. Since K ⊂ K for every K ∈ P max , this shows that (K, ρ) is rigid in the sense of Definition 3.9.

Corollary 5.2 A Moy-Prasad unrefined minimal K -type of positive level is rigid.
Proof This is immediate from Proposition 5.1.
By [15,Proposition 5.3], every Moy-Prasad unrefined minimal K -type of positive level is a Bushnell-Kutzko type. This proves Theorem 1.2 for irreducible G-representations which contain a Moy-Prasad type of positive level. By [35,Theorem 5.2], if that is not the case, then the irreducible Grepresentation contains a level zero type.

Level zero types
We now look at the case of level zero types [32][33][34]. We refer to the loc. cit. for the necessary structural results from Bruhat-Tits theory, in particular, about parahoric subgroups. Fix an Iwahori subgroup I , and recall the Bruhat decomposition where W is the Iwahori-Weyl group andẇ is a representative in G of w. Let be the set of affine simple roots defined by I and W the affine Weyl group. We have W = W , where can be identified with the subgroup of W of elements of length zero. Let P ⊇ I be a parahoric subgroup. It corresponds to a subset J and we write P = P J to emphasize this relation. Let W J denote the (finite) subgroup of W generated by the reflections s α , α ∈ J . There is a one-to-one correspondence W J \W/W J ←→ P J \G/P J , W J wW J → P Jẇ P J = ∪ u∈W J wW J Iu I.  The set W (J ) is a subgroup of W . (P, ρ), where P = P J is a parahoric subgroup and ρ is a representation of P J inflated from a cuspidal representation of the finite reductive quotient M J of P J .

Definition 5.3 A level zero type is a pair
The main result of [33] is a description of the ρ-spherical Hecke algebra H (G, ρ). The starting essential observation [33,Theorem 4.15] is that a coset Pẇ P, with w of minimal length in W J wW J , supports a nonzero element of H(G, ρ) only if w ∈ N W (W J ) and w ρ ∼ = ρ. This is a compact open subgroup of G and P J ⊂ K J , P Jẇ P J ⊂ K J . This proves:
T wv a , if w(a) > 0, p a T wv a + ( p a − 1)T w , if w(a) < 0.
Here p a = 1 are certain nonnegative powers of the residual characteristic.
In other words, H(G, ρ) is the smash-product of an affine Hecke algebra with a twisted group algebra. This means that classification of unitary representations of level zero of the p-adic group G is equivalent with the classification of the unitary dual of the Hecke algebras in Theorem 5.5. When G is adjoint and ρ is unipotent, Lusztig [32] gives a complete description of H(G, ρ). In that case, the cocycle μ is trivial, and H(G, ρ) is an affine Hecke algebra with unequal parameters.
Remark 5.6 When the cocycle μ is trivial, an explicit description of the cocenter and compact cocenter of H(G, ρ) is available by [18,25].

Inner forms of GL(n)
Suppose now that G = G L(n, F) or one of its inner forms and that (K, ρ) is a type of G. In this case, there exists a unique conjugacy class of maximal compact open subgroups. Let K 0 denote a representative of the class. Without loss of generality, we may assume that K ⊆ K 0 . Since every element in H c (G) is represented by a function supported on K 0 , (K, ρ) is trivially rigid.  L(m, D), where D is a d 2 -dimensional central division F-algebra, n = md, we know a complete list of s-types, s ∈ B(G), by the work of [13] (for G L(n, F)) and [41] (in general). All of these s-types are rigid by the previous paragraph.