3.1 Hecke algebra

Our reference here is [Reference Bushnell and Kutzko19, §4] and [Reference Bushnell and Kutzko22, §2]. Let G be an $\ell $ -group, let J be an open compact subgroup of G, and let $(\rho ,W)$ be an irreducible representation of J, which is necessarily finite dimensional. We fix a Haar measure $dx$ of G.

Let $\mathcal {H}(G)$ denote the Hecke algebra of G consisting of complex smooth compactly supported functions on G, equipped with the convolution for the product structure. It is well-known that we have an equivalence of categories

$$ \begin{align*}\operatorname{Rep}(G)\rightarrow\operatorname{Mod}(\mathcal{H}(G)),\end{align*} $$

where $\operatorname {Mod}(\mathcal {H}(G))$ denotes the category of nondegenerate $\mathcal {H}(G)$ -modules, and for $(\pi ,V)\in \operatorname {Rep}(G)$ , the corresponding $\mathcal {H}(G)$ -module structure is given by

$$ \begin{align*}v\mapsto \pi(f)v:=\int_{G}f(x)\pi(x)vdx\end{align*} $$

for $f\in \mathcal {H}(G)$ and $v\in V$ .

We denote by $\mathcal {H}(G,\rho )$ the space of compactly supported functions $\phi :G\rightarrow \operatorname {End}_{\mathbb {C}}(W^{\vee })$ , such that

$$ \begin{align*}\phi(g_{1}gg_{2})=\rho^{\vee}(g_{1})\phi(g)\rho^{\vee}(g_{2}),\quad g_{1},g_{2}\in J,\ g\in G.\end{align*} $$

The convolution operation

$$ \begin{align*}\phi_{1}\ast \phi_{2}(g)=\int_{G}\phi_{1}(x)\phi_{2}(x^{-1}g)dx,\quad \phi_{1},\phi_{2}\in\mathcal{H}(G,\rho)\end{align*} $$

gives $\mathcal {H}(G,\rho )$ the structure of an associative $\mathbb {C}$ -algebra with unit. We will use the abbreviation

$$ \begin{align*}\phi^{k}=\underbrace{\phi\ast\dots\ast\phi}_{k\text{-copies}}.\end{align*} $$

By [Reference Bushnell and Kutzko19, Proposition 4.1.1], for $g\in G$ , there exists a function $\phi \in \mathcal {H}(G,\rho )$ whose support is $JgJ$ if and only if g intertwines $\rho $ .

We fix a complex structure on the finite dimensional complex vector space $W^{\vee }$ . So $(\rho ^{\vee },W^{\vee })$ is realized as a unitary representation. We define

$$ \begin{align*}h_{G}(\phi_{1},\phi_{2})=\int_{x\in G}\operatorname{tr}_{W^{\vee}}(\phi_{1}(x)\overline{\phi_{2}(x)})dx=\operatorname{tr}_{W^{\vee}}(\phi_{1}\ast\overline{\phi_{2}}(1)),\quad \phi_{1},\phi_{2}\in\mathcal{H}(G,\rho)\end{align*} $$

as a positive definite hermitian form on $\mathcal {H}(G,\rho )$ . Here, the ‘bar’ for $\overline {\phi _{2}(x)}$ denotes the complex conjugate on $\operatorname {End}_{\mathbb {C}}(W^{\vee })$ induced by that on $W^{\vee }$ and the ‘bar’ for $\overline {\phi _{2}}$ is given by $\overline {\phi _{2}}(x)=\overline {\phi _{2}(x^{-1})}$ for $x\in G$ , and $\operatorname {tr}_{W^{\vee }}:\operatorname {End}(W^{\vee })\rightarrow \mathbb {C}$ denotes the trace map. Of course, $h_{G}$ depends on the choice of the Haar measure $dx$ on G.

For $a\in \operatorname {End}_{\mathbb {C}}(W)$ , we write $a^{\vee }\in \operatorname {End}_{\mathbb {C}}(W^{\vee })$ for the transpose of a with respect to the canonical pairing between W and $W^{\vee }$ . Then we get an isomorphism of algebras

$$ \begin{align*}\mathcal{H}(G,\rho)\rightarrow\mathcal{H}(G,\rho^{\vee})^{{op}},\quad \phi\mapsto \hat{\phi}:=[g\mapsto \phi(g^{-1})^{\vee}].\end{align*} $$

The map

$$ \begin{align*}(\phi,f)\mapsto\phi\ast f:=[g\mapsto\int_{G}\phi(x)f(x^{-1}g)],\quad \phi\in\mathcal{H}(G,\rho^{\vee}),\ f\in\operatorname{ind}_{J}^{G}\rho\end{align*} $$

induces an isomorphism of algebras $\mathcal {H}(G,\rho ^{\vee })\cong \operatorname {End}_{G}(\operatorname {ind}_{J}^{G}\rho ).$ Also, it provides $\operatorname {ind}_{J}^{G}\rho $ with a left $\mathcal {H}(G,\rho ^{\vee })$ -module structure and thus a right $\mathcal {H}(G,\rho )$ -module structure.

Let $(\pi ,V)\in \operatorname {Rep}(G)$ . Let

$$ \begin{align*}V_{\rho}:=\operatorname{Hom}_{J}(W,V)=\operatorname{Hom}_{G}(\operatorname{ind}_{J}^{G}W,V)\end{align*} $$

be the space of $\rho $ -invariants of $(\pi ,V)$ . It is endowed with a left $\mathcal {H}(G,\rho )$ -module structure via the right $\mathcal {H}(G,\rho )$ -action on $\operatorname {ind}_{J}^{G}\rho $ explained as above. More precisely, the action is given by

(3.1) $$ \begin{align} \phi\cdot f=[w\mapsto\int_{G}\pi(g)f(\hat{\phi}(g^{-1})w)dg],\quad \phi\in\mathcal{H}(G,\rho),\ f\in V_{\rho},\ w\in W. \end{align} $$

This provides us with a functor

(3.2) $$ \begin{align} \boldsymbol{\mathrm{M}}_{\rho}:\operatorname{Rep}(G)\rightarrow\operatorname{Mod}(\mathcal{H}(G,\rho)),\quad (\pi,V)\mapsto V_{\rho}. \end{align} $$

Here, $\operatorname {Mod}(\mathcal {H}(G,\rho ))$ denotes the equivalence classes of nondegenerate $\mathcal {H}(G,\rho )$ -modules.

3.2 Cuspidal representations

In the rest of this section, let G be a p-adic reductive group or, more generally, an n-fold cover of a p-adic reductive group (cf. [Reference Renard60], [Reference Kaplan and Szpruch47]).

Let $P=MN$ be a parabolic subgroup of G, with M being a Levi factor and N being the unipotent radical of P.

We define, as in [Reference Bernstein and Zelevinsky6, §1.8], the normalized parabolic induction functor and normalized Jacquet functor

$$ \begin{align*}i_{P}^{G}:\operatorname{Rep}(M)\rightarrow\operatorname{Rep}(G),\quad r_{N}:\operatorname{Rep}(G)\rightarrow\operatorname{Rep}(M).\end{align*} $$

We recall the following equivalent definitions for a cuspidal representation of G.

One typical method of obtaining cuspidal representations is the usage of compact induction, which is summarized as the following lemma:

3.3 Bernstein blocks and types

A cuspidal pair $(M,\rho )$ of G consists of a Levi subgroup M of G and a cuspidal representation $\rho $ of M. It is known that for an irreducible representation $\pi $ of G, there exists a cuspidal pair $(M,\rho )$ , unique up to G-conjugacy, and a parabolic subgroup P of G having a Levi factor M, such that $\pi $ is a subrepresentation of the parabolic induction $i_{P}^{G}(\rho )$ , where P is a parabolic subgroup of M. Such a pair is called the cuspidal support of $\pi $ .

Moreover, let $\mathcal {O}$ be the orbit of $\rho $ twisted by unramified characters of M. Then $\mathfrak {s}=(M,\mathcal {O})$ , up to G-conjugacy, is called an inertial equivalence class of G. The inertial support of $\pi $ is the inertial equivalence class of its cuspidal support.

Write $\operatorname {Rep}_{\mathfrak {s}}(G)$ for the subcategory of $\operatorname {Rep}(G)$ consisting of representations whose irreducible subquotients have inertial support $\mathfrak {s}$ . Then, each $\operatorname {Rep}_{\mathfrak {s}}(G)$ is a block of $\operatorname {Rep}(G)$ , and we have the block decomposition (cf. [Reference Renard60, Chapitre IV, §7.2], [Reference Kaplan and Szpruch47])

$$ \begin{align*}\operatorname{Rep}(G)\cong\prod_{\mathfrak{s}}\operatorname{Rep}_{\mathfrak{s}}(G),\end{align*} $$

where $\mathfrak {s}$ ranges over all the inertial equivalence classes of G.

Let J be an open compact subgroup of G and let $\lambda $ be an irreducible representation of J. As in [Reference Bushnell and Kutzko22, §3], we may define a special idempotent $e_{\lambda }$ in $\mathcal {H}(G)$ and the subcategory $\operatorname {Rep}_{\lambda }(G)=\{V\mid \mathcal {H}(G)e_{\lambda } V=V\}$ of $\operatorname {Rep}(G)$ . Then, we call $(J,\lambda )$ an $\mathfrak {s}$ -type if for any irreducible representation $\pi $ of G, we have that $\pi |_{J}$ contains $\lambda $ if and only if $\pi $ has inertial equivalence class $\mathfrak {s}$ .

In this case, we have $\operatorname {Rep}_{\lambda }(G)=\operatorname {Rep}_{\mathfrak {s}}(G)$ and an equivalence of categories

$$ \begin{align*}\boldsymbol{\mathrm{M}}_{\lambda}:\operatorname{Rep}_{\mathfrak{s}}(G)\rightarrow\operatorname{Mod}(\mathcal{H}(G,\lambda)),\quad (\pi,V)\mapsto V_{\lambda}.\end{align*} $$

Such $\boldsymbol {\mathrm {M}}_{\lambda }$ is equivariant up to the multiplicative action by unramified characters of G on both $\operatorname {Rep}_{\mathfrak {s}}(G)$ and $\operatorname {Mod}(\mathcal {H}(G,\lambda ))$ .

3.4 Covering pairs

Let $P=MN$ be a parabolic subgroup of G, with M being a Levi factor and N being the unipotent radical of P. Let $P^{-}$ and $N^{-}$ be the opposite of P and N, respectively, with respect to M. A pair $(J,\lambda )$ , consisting of an open subgroup J of G and an irreducible $\lambda $ of J is called decomposed with respect to $(M,P)$ if

• • $J=(J\cap N^{-})\cdot (J\cap M)\cdot (J\cap N)$ .

• • The groups $J\cap N^{-}$ and $J\cap N$ are contained in the kernel of $\lambda $ .

An element $\zeta \in M$ is called strongly $(P,J)$ -positive if

• • $\zeta (J\cap N)\zeta ^{-1}\subset (J\cap N)$ , $\zeta ^{-1}(J\cap N^{-})\zeta \subset (J\cap N^{-}).$

• • For any compact open subgroups $H_{1}$ , $H_{2}$ of N, there exists an integer $i\geq 0$ such that $\zeta ^{i}H_{1}\zeta ^{-i}\subset H_{2}$ .

• • For any compact open subgroups $H_{1}$ , $H_{2}$ of $N^{-}$ , there exists an integer $i\geq 0$ such that $\zeta ^{-i}H_{1}\zeta ^{i}\subset H_{2}$ .

Consider another pair $(J_{M},\lambda _{M})$ , where $J_{M}$ is an open compact subgroup of M and $\lambda _{M}$ is an irreducible representation of $J_{M}$ . The pair $(J,\lambda )$ is a (G-)covering pair of $(J_{M},\lambda _{M})$ if

1. (1) The pair $(J,\lambda )$ is decomposed with respect to $(M,P)$ for every parabolic subgroup P of G with a Levi factor M.

2. (2) $J\cap M=J_{M}$ and $\lambda |_{J_{M}}=\lambda _{M}$ .

3. (3) For every parabolic subgroup P of G with a Levi factor M, there exist a strongly $(P,J)$ -positive element $z\in Z(M)$ and an invertible element $\phi _{z}$ in $\mathcal {H}(G,\lambda )$ supported on $JzJ$ .

Sometimes, the following condition is more convenient to verify, which implies the condition (3) above (cf. [Reference Bushnell and Kutzko22, Condition 8.2]):

1. (4) Every $\phi \in \mathcal {H}(G,\lambda )$ is supported on $JMJ$ .

We remark that in [Reference Bushnell and Kutzko22], Bushnell and Kutzko use the word ‘cover’ for such a pair $(J,\lambda )$ , which unfortunately conflicts with ‘metaplectic covers’, the central objects of this article. So we use ‘covering pair’ instead, translated from the corresponding French words (cf. [Reference Blondel10]).

The following theorem illustrates the connection between types and covering pairs.

Fix a parabolic subgroup $P=MN$ of G. Let $(J,\lambda )$ be a covering pair of $(J_{M},\lambda _{M})$ . In [Reference Bushnell and Kutzko22, §7], an embedding of algebras $t_{P}:\mathcal {H}(M,\lambda _{M})\rightarrow \mathcal {H}(G,\lambda )$ is defined and uniquely characterized by the following properties:

• • For any strongly $(P,J)$ -positive element $\zeta \in M$ , define $\varphi _{\zeta }\in \mathcal {H}(M,\lambda _{M})$ (resp. $\phi _{\zeta }\in \mathcal {H}(G,\lambda )$ ) such that $\operatorname {Supp}(\varphi _{\zeta })=J_{M}\zeta J_{M}$ and $\varphi _{\zeta }(\zeta )=\mathrm {id}_{\lambda _{M}^{\vee }}$ (resp. $\operatorname {Supp}(\phi _{\zeta })= J\zeta J$ and $\phi _{\zeta }(\zeta )=\mathrm {id}_{\lambda ^{\vee }}).$ Then $t_{P}(\varphi _{\zeta })=\delta _{N}^{1/2}(\zeta )\cdot \phi _{\zeta }$ , where $\delta _{N}:M\rightarrow \mathbb {C}^{\times }$ denotes the modulus character with respect to N.

• • Let $\mu _{G}$ (resp. $\mu _{M}$ ) be the Haar measure on G (resp. M), such that $\mu _{G}(J)=\mu _{M}(J_{M})=1$ . Using these two Haar measures and fixing a complex structure on the representation space of $\lambda ^{\vee }$ and $\lambda _{M}^{\vee }$ , we define the corresponding hermitian forms $h_{G}$ on $\mathcal {H}(G,\lambda )$ and $h_{M}$ on $\mathcal {H}(M,\lambda _{M})$ . Then we have

  (3.3) $$ \begin{align} h_{M}(\varphi,\varphi)=h_{G}(t_{P}(\varphi),t_{P}(\varphi)),\quad\text{for any}\ \varphi\in\mathcal{H}(M,\lambda_{M}). \end{align} $$
  In other words, $t_{P}$ is an isometry between $\mathcal {H}(M,\lambda _{M})$ and $t_{P}(\mathcal {H}(G,\lambda ))$ .

• • If $\mathcal {H}(G,\lambda )_M:=\{\phi \in \mathcal {H}(G,\lambda )\mid \operatorname {Supp}(\phi )\subset JMJ \}$ is a subalgebra of $\mathcal {H}(G,\lambda )$ , then we have $t_P(\mathcal {H}(M,\lambda _M))=\mathcal {H}(G,\lambda )_M$ and the map $t_P$ preserves support of functions (cf. [Reference Bushnell and Kutzko22, Theorem 7.2.(ii)]), in the sense that

  $$ \begin{align*}\operatorname{Supp}(t_{P}(\varphi))=J\cdot\operatorname{Supp}(\varphi)\cdot J,\quad \varphi\in\mathcal{H}(M,\lambda_{M}).\end{align*} $$

We remark that the second property follows from [Reference Roche61, §5]. In loc. cit. the condition (4) for a covering pair is imposed, which simply guarantees that $t_{P}$ is an isomorphism and is not essential for our statement here. Moreover, although in loc. cit. the representation $\lambda $ is assumed to be one-dimensional, as already pointed out by the author, the proof is general enough to be adapted to an irreducible finite dimensional representation $\lambda $ of J.

The above homomorphism $t_{P}$ induces a map

$$ \begin{align*}t_{P}^{*}:\operatorname{Mod}(\mathcal{H}(G,\lambda))\rightarrow\operatorname{Mod}(\mathcal{H}(M,\lambda_{M}))\end{align*} $$

as the pull-back map of $t_{P}$ . Such $t_{P}^{*}$ should be regarded as the normalized Jacquet functor on the Hecke algebra side. Also, the above $t_{P}^{*}$ has a right adjoint

$$ \begin{align*}(t_{P})_{*}:\operatorname{Mod}(\mathcal{H}(M,\lambda_{M}))\rightarrow \operatorname{Mod}(\mathcal{H}(G,\lambda)).\end{align*} $$

Indeed, for $V\in \operatorname {Mod}(\mathcal {H}(M,\lambda _{M}))$ , we have

$$ \begin{align*}(t_{P})_{*}(V)=\operatorname{Hom}_{\mathcal{H}(M,\lambda_{M})}(\mathcal{H}(G,\lambda),V).\end{align*} $$

Correspondingly, such $(t_{P})_{*}$ should be regarded as the normalized parabolic induction functor on the Hecke algebra side. More precisely, we have the following theorem:

Theorem 3.4 ([Reference Bushnell and Kutzko22, Theorem 7.9, Corollary 8.4]).

Under the setting of Theorem 3.3, we have commutative diagrams:

4.1 General theory for a finite central cover

We refer to [Reference Gan, Gao and Weissman29] for the basic facts that we are going to state below, which also serves as an excellent historical survey.

Let G be an $\ell $ -group. By an n-fold cover of G, we mean a central extension of G by $\mu _{n}$ as $\ell $ -groups:

(4.1)

We note that the set of equivalence classes of such central extensions is in bijection with the set of (locally constant) 2-cohomology classes $H^{2}(G,\mu _{n})$ . More precisely, given an n-fold cover $\widetilde {G}$ of G, we fix a continuous map $\boldsymbol {s}:G\rightarrow \widetilde {G}$ such that $\boldsymbol {p}\circ \boldsymbol {s}=\operatorname {id}$ . Then the corresponding cohomology class is related to the 2-cocycle $\sigma :G\times G\rightarrow \mu _{n}$ satisfying

(4.2) $$ \begin{align} \boldsymbol{s}(g_{1})\boldsymbol{s}(g_{2})\sigma(g_{1},g_{2})=\boldsymbol{s}(g_{1}g_{2}),\quad g_{1},g_{2}\in G. \end{align} $$

For two covers $\widetilde {G}_{1}$ and $\widetilde {G}_{2}$ realized by 2-cocycles $\sigma _{1}$ and $\sigma _{2}$ , respectively, the Baer sum of $\widetilde {G}_{1}$ and $\widetilde {G}_{2}$ is realized by the 2-cocycle $\sigma _{1}\cdot \sigma _{2}$ .

By convention, for a closed subgroup H of G, we write $\widetilde {H}$ for the preimage $\boldsymbol {p}^{-1}(H)$ . We identify $\mu _{n}$ with a central subgroup of $\widetilde {G}$ .

A splitting of H is a continuous group homomorphism $\boldsymbol {s}_{H}:H\rightarrow \widetilde {G}$ satisfying $\boldsymbol {p}\circ \boldsymbol {s}_{H}=\operatorname {id}$ . In general, such a splitting may not exist or may not be unique. If H is a pro-p-group and $\operatorname {gcd}(n,p)=1$ , then the cohomology group $H^{2}(H,\mu _{n})$ is trivial, implying that there exists a unique splitting $\boldsymbol {s}_{H}$ of H. Thus, we identify H with a subgroup $\boldsymbol {s}_{H}(H)$ of $\widetilde {H}$ , which we denote by $\,_{s}H$ . For a representation $\rho $ of H, we similarly write $\,_{s}\rho $ for the corresponding representation of $\,_{s}H$ . In general, even if the splittings of H are not unique, we may still define $\,_{s}H$ and $\,_{s}\rho $ as above once we fix a splitting $\boldsymbol {s}_{H}$ . We refer to [Reference Ngo58] for a similar setting.

The commutator $[\cdot ,\cdot ]:\widetilde {G}\times \widetilde {G}\rightarrow \widetilde {G}$ factors through $G\times G$ , and we denote by $[\cdot ,\cdot ]_{\sim }:G\times G\rightarrow \widetilde {G}$ the resulting map. In particular, if $g_{1},g_{2}\in G$ commute, then

(4.3) $$ \begin{align} [g_{1},g_{2}]_{\sim}=\sigma(g_{1},g_{2})\sigma(g_{2},g_{1})^{-1}\in\mu_{n}. \end{align} $$

Similarly, the $\widetilde {G}$ -conjugation on $\widetilde {G}$ factors through G, so we may consider the G-conjugation on $\widetilde {G}$ . It is clear that $[g_{1}^{g},g_{2}^{g}]_{\sim }=[g_{1},g_{2}]_{\sim }$ for any $g,g_{1},g_{2}\in G$ . Also, for $H,H_{1},H_{2}$ closed subgroups of G, we define the coset $g\widetilde {H}:=\boldsymbol {s}(g)\widetilde {H}$ and the double coset $\widetilde {H_{1}}g\widetilde {H_{2}}:=\widetilde {H_{1}}\boldsymbol {s}(g)\widetilde {H_{2}}$ , which does not depend on the choice of the section $\boldsymbol {s}$ .

For $\widetilde {\rho }\in \operatorname {Rep}(\widetilde {H})$ , we say that $g\in G$ intertwines $\widetilde {\rho }$ if $\boldsymbol {s}(g)$ intertwines $\widetilde {\rho }$ , which does not depend on the choice of $\boldsymbol {s}$ . We denote by $I_{G}(\widetilde {\rho })$ the intertwining set of $\widetilde {\rho }$ as a subset of G. Similarly, it makes sense to define the normalizer $N_{G}(\widetilde {\rho })$ .

Fix a character $\epsilon :\mu _{n}\rightarrow \mathbb {C}^{\times }$ . A representation $\widetilde {\pi }$ of $\widetilde {G}$ is called ( $\epsilon $ -)genuine if $\mu _{n}$ acts by $\epsilon $ on $\widetilde {\pi }$ . We remark that essentially we only need to consider the case where $\epsilon $ is primitive; otherwise, any $\epsilon $ -genuine representation $\widetilde {\pi }$ of $\widetilde {G}$ corresponds to an $\epsilon '$ -genuine representation of an $n'$ -fold cover of G with $n'$ dividing n, where $\epsilon '$ is a primitive character of $\mu _{n'}$ . So from now on, we also assume the character $\epsilon $ to be primitive.

For a representation $\rho $ and a splitting $\boldsymbol {s}$ of H, we write $\epsilon \cdot \,_{s}\rho $ for the extension of $\,_{s}\rho $ as an $\epsilon $ -genuine representation of $\widetilde {H}$ .

For a representation $\chi $ of H and a genuine representation $\widetilde {\rho }$ of $\widetilde {H}$ , the tensor product $\widetilde {\rho }\otimes \chi :=\widetilde {\rho }\otimes (\chi \circ \boldsymbol {p})$ is well-defined as another genuine representation of $\widetilde {H}$ . When $\chi $ is a character, we write $\widetilde {\rho }\chi $ or $\widetilde {\rho }\cdot \chi $ instead.

4.2 Metaplectic covers of $\operatorname {GL}_{r}(F)$

We consider metaplectic covers of $\operatorname {GL}_{r}(F)$ , which, in the context of this article, are n-fold covers arising from Brylinski-Deligne covers (cf. [Reference Brylinski and Deligne14]). Roughly speaking, Brylinski and Deligne constructed a family of central extensions of a reductive group G over F by the second K-group:

(4.4)

Recall that the abelian group $K_{2}(F)$ could be given by generators of the form $\{x,y\}$ with $x,y\in F^{\times }$ , subject only to the following relations (cf. [Reference Matsumoto51], [Reference Milnor52, Theorem 11.1]):

• • $\{x_{1}x_{2},y\}=\{x_{1},y\}\{x_{2},y\},\ \{y,x_{1}x_{2}\}=\{y,x_{1}\}\{y,x_{2}\}, \quad \text {for any}\ x_{1},x_{2},y\in F^{\times }.$

• • $\{x,1-x\}=1,\quad \text {for any}\ x\neq 0,1.$

Pushing out the map (cf. [Reference Gan, Gao and Weissman29, p3])

$$ \begin{align*}K_{2}(F)\rightarrow \mu_{n},\quad \{x,y\}\mapsto (x,y)_{n},\end{align*} $$

they got a family of n-fold covers $\widetilde {G}$ of G.

We follow [Reference F. Gao and Weissman30] to characterize all the metaplectic covers of $G=\operatorname {GL}_{r}(F)$ . Let T be the diagonal torus of G, let $Y=\operatorname {Hom}(\mathbb {G}_{m},T)$ be the cocharacter lattice, and let $W=W(G,T)$ be the corresponding Weyl group. Then the set of $K_{2}(F)$ -extensions $\hat {G}$ of G are in bijection with the set of W-invariant quadratic forms

$$ \begin{align*}Q:Y\rightarrow \mathbb{Z}.\end{align*} $$

This bijection is additive, saying that for two Brylinski-Deligne covers $\hat {G}_{1}$ and $\hat {G}_{2}$ of G with related quadratic forms being $Q_{1}$ and $Q_{2}$ , respectively, the quadratic form related to the Baer sum of $\hat {G}_{1}$ and $\hat {G}_{2}$ is $Q_{1}+Q_{2}$ .

One way to pin down the above bijection is as follows: Let $B_{Q}$ be the associated bilinear form given by

$$ \begin{align*}B_{Q}(y_{1},y_{2})=Q(y_{1}+y_{2})-Q(y_{1})-Q(y_{2}),\quad y_{1},y_{2}\in Y.\end{align*} $$

Then for $y_{1},y_{2}\in Y$ and $a,b\in F^{\times }$ , we have (cf. [Reference Brylinski and Deligne14, Corollary 3.14])

$$ \begin{align*}[y_{1}(a),y_{2}(b)]_{\sim}=\{a,b\}^{B_{Q}(y_{1},y_{2})},\end{align*} $$

where $[\cdot ,\cdot ]_{\sim }:T\times T\rightarrow K_{2}(F)$ is the resulting commutator with respect to $\hat {G}$ .

Moreover, we let $e_{1},\dots ,e_{r}$ be the canonical basis of Y. Then such quadratic forms are in bijection with pairs of integers $(\boldsymbol {a},\boldsymbol {b})$ given by $\boldsymbol {a}=Q(e_{1})$ and $\boldsymbol {b}=B_{Q}(e_{1},e_{2})$ . Likewise, this bijection is additive.

We consider two special Brylinski-Deligne covers of G. First, we consider the $K_{2}(F)$ -extension $\hat {G}_{1}$ related to the 2-cocycle

$$ \begin{align*}G\times G\rightarrow K_{2}(F),\quad (g_{1},g_{2})\mapsto\{\operatorname{det}(g_{1}),\operatorname{det}(g_{2})\}.\end{align*} $$

In this case, the corresponding pair of integers is $(\boldsymbol {a},\boldsymbol {b})=(1,2)$ . We let $\widetilde {G}_{1}$ be the corresponding n-fold metaplectic cover of G. Then it is related to the 2-cocycle

(4.5) $$ \begin{align} \sigma_{\operatorname{det}}:G\times G\rightarrow\mu_{n},\quad (g_{1},g_{2})\mapsto(\operatorname{det}(g_{1}),\operatorname{det}(g_{2}))_{n}. \end{align} $$

We further consider the canonical $K_{2}(F)$ -extension of $\operatorname {SL}_{r+1}(F)$ considered by Matsumoto with respect to the Steinberg symbol $\{\cdot ,\cdot \}^{-1}:F^{\times }\times F^{\times }\rightarrow K_{2}(F)$ (cf. [Reference Matsumoto51], [Reference Milnor52, §12]). Via the pull-back of the embedding

$$ \begin{align*}\operatorname{GL}_{r}(F)\rightarrow\operatorname{SL}_{r+1}(F),\quad g\mapsto(\operatorname{det}(g)^{-1},g),\end{align*} $$

we get a $K_{2}(F)$ -extension of G, which we denote by $\hat {G}_{2}$ . In this case, the corresponding pair of integers is $(\boldsymbol {a},\boldsymbol {b})=(1,1)$ . We let $\widetilde {G}_{2}$ be the corresponding n-fold metaplectic cover of $G=\operatorname {GL}_{r}(F)$ . In [Reference Kazhdan and Patterson48, §0.I], [Reference Banks, Levy and Sepanski4], a special 2-cocycle $\sigma _{\mathrm {KP}}:G\times G\rightarrow \mu _{n}$ related to the cover $\widetilde {G}_{2}$ is constructed for each r. It is trivial when $r=1$ and satisfies the following block-compatibility:

(4.6) $$ \begin{align} \begin{aligned} \sigma_{\mathrm{KP}}(\operatorname{diag}(g_1,\dots,g_k),\operatorname{diag}(g_1',\dots,g_k'))= \bigg[\prod_{i=1}^k\sigma_{\mathrm{KP}}(g_i,g_i')\bigg]\cdot\bigg[\prod_{1\leq i<j\leq k}(\det(g_i),\det(g_j'))_n\bigg], \end{aligned} \end{align} $$

where $g_{i},g_{i}'\in \operatorname {GL}_{r_{i}}(F)$ for each $i=1,\dots ,k$ and $r=r_{1}+\cdots +r_{k}$ .

Since $(1,2)$ and $(1,1)$ generate $\mathbb {Z}\times \mathbb {Z}$ , the integral combination of $\hat {G}_{1}$ and $\hat {G}_{2}$ with respect to the Baer sum ranges over all the equivalence classes of Brylinski-Deligne covers of G, and the integral combination of $\widetilde {G}_{1}$ and $\widetilde {G}_{2}$ ranges over all the n-fold metaplectic covers of G. In other words, when $(\boldsymbol {c},\boldsymbol {d})$ ranges over $\mathbb {Z}/n\mathbb {Z}\times \mathbb {Z}/n\mathbb {Z}$ , the corresponding 2-cocycle $\sigma _{\operatorname {det}}^{\boldsymbol {c}}\cdot \sigma _{\mathrm {KP}}^{\boldsymbol {d}}$ represents all the n-fold metaplectic covers of G.

We fix a pair of integers $(\boldsymbol {c},\boldsymbol {d})$ , and we let $\sigma =\sigma _{\operatorname {det}}^{\boldsymbol {c}}\cdot \sigma _{\mathrm {KP}}^{\boldsymbol {d}}$ be the related 2-cocycle for $r\geq 1$ . We let $\widetilde {G}$ be the related n-fold metaplectic cover of G. We list various properties of $\sigma $ .

Proof. Statement (1) is direct.

Statement (2)(3)(4) follow from [Reference Kaplan, Lapid and Zou46, Lemma 4.1], and statement (5a) follows from [Reference Kazhdan and Patterson48, Proposition 0.1.5] in the case $\boldsymbol {d}=1$ . The general cases follow easily from the fact that $\sigma =\sigma _{\mathrm {KP}}^{\boldsymbol {d}}\cdot \sigma _{\operatorname {det}}^{\boldsymbol {c}}$ and a direct calculation (cf. (4.3)).

For statement (5b), we notice that $[u,\cdot ]_{\sim }$ is a character of $\operatorname {GL}_{r_{1}'}(E_{1})\times \dots \times \operatorname {GL}_{r_{k}'}(E_{k})$ , which is trivial on the derived subgroup $\operatorname {SL}_{r_{1}'}(E_{1})\times \dots \times \operatorname {SL}_{r_{k}'}(E_{k})$ . So we only need to verify the formula (4.11) for those $v_{i}$ diagonal in $\operatorname {GL}_{r_{i}'}(E_{i})$ for each i. Then we may use (4.10) with respect to an F-algebra embedding

$$ \begin{align*}\underbrace{E_{1}\times\dots\times E_{1}}_{r_{1}'\text{-copies}}\times\dots\times \underbrace{E_{k}\times\dots\times E_{k}}_{r_{k}'\text{-copies}}\hookrightarrow \operatorname{M}_{r}(F).\end{align*} $$

The first part of statement (6) follows from [Reference Kazhdan and Patterson48, Proposition 0.1.2] and the fact that $(\cdot ,\cdot )_{n,F}$ is trivial on $\mathfrak {o}_{F}^{\times }\times \mathfrak {o}_{F}^{\times }$ when $\operatorname {gcd}(q,n)=1$ . The second part is trivial. In the final part, since K and $\operatorname {GL}_{r}(\mathfrak {o}_{F})$ are conjugate by some $g\in G$ , we simply take the conjugation of the corresponding splitting of $\operatorname {GL}_{r}(\mathfrak {o}_{F})$ as a splitting of K.

From now on, we fix $r\geq 1$ , $(\boldsymbol {c},\boldsymbol {d})\in \mathbb {Z}\times \mathbb {Z}$ and $\sigma $ as above. We let $\widetilde {G}$ be the n-fold metaplectic cover of $G=\operatorname {GL}_{r}(F)$ corresponding to $\sigma $ . For a closed subgroup H of G, we define

$$ \begin{align*}H^{(n)}=\{h\in H\mid\operatorname{det}(h)\in F^{\times n}\}.\end{align*} $$

Let $P=MN$ be a parabolic subgroup of G with a Levi factor M and the unipotent radical N, where we assume that M is isomorphic to $\operatorname {GL}_{r_{1}}(F)\times \dots \times \operatorname {GL}_{r_{k}}(F)$ with $r=r_{1}+\dots +r_{k}$ . Since N is a pro-p-group, there exists a unique splitting of N into $\widetilde {G}$ . Thus, we may realize N as a subgroup of $\widetilde {G}$ , which we still denote by N. So $\widetilde {P}=\widetilde {M}N$ is a parabolic subgroup of $\widetilde {G}$ with a Levi factor $\widetilde {M}$ and the unipotent radical N.

Let $H=H_{1}\times \dots \times H_{k}$ be a subgroup of M, such that each $H_{i}$ is a closed subgroup of $\operatorname {GL}_{r_{i}}(F)$ . We call H block compatible if $[H_{i},H_{j}]_{\sim }=\{1\}$ for any $1\leq i<j\leq k$ . We notice that H is block compatible in the following two cases (cf. (4.8)):

• • For any i, we have $H_{i}=H_{i}^{(n)}$ ;

• • For any i, the determinant of every element in $H_{i}$ is in $\mathfrak {o}_{F}^{\times }$ .

This concept becomes important when considering (exterior) tensor product of genuine representations. Let $\widetilde {\rho }_{i}$ be genuine representations of $\widetilde {H_{i}}$ for each i. We take the tensor product $\widetilde {\rho }_{1}\boxtimes \dots \boxtimes \widetilde {\rho }_{k}$ as a representation of $\widetilde {H_{1}}\times \dots \times \widetilde {H_{k}}$ , which is trivial on

$$ \begin{align*}\Xi=\{(\zeta_{1},\dots,\zeta_{k})\in\mu_{n}\times\dots\times\mu_{n}\mid \zeta_{1}\dots\zeta_{k}=1\}.\end{align*} $$

If H is block compatible, then it is clear that

$$ \begin{align*}\widetilde{H}\cong\widetilde{H_{1}}\times\dots\times\widetilde{H_{k}}/\Xi,\end{align*} $$

so we realize $\widetilde {\rho }_{1}\boxtimes \dots \boxtimes \widetilde {\rho }_{k}$ as a genuine representation of $\widetilde {H}$ .

In particular, we emphasize the following three special classes of covers:

• • (Determinantal covers) When $\boldsymbol {d}=0$ , we get determinantal covers. We remark that such a cover is not far from the corresponding linear group since for any closed subgroup H of G, the group $\widetilde {H^{(n)}}$ is isomorphic to $H^{(n)}\times \mu _{n}$ . Moreover, $H/H^{(n)}$ is of finite index. So the representation theory of $\widetilde {G}$ is easily deduced from that of G.

• • (The Kazhdan-Patterson covers) When $\boldsymbol {d}=1$ , we get Kazhdan-Patterson covers (KP-covers for short) (cf. [Reference Kazhdan and Patterson48]) Such covers are natural, in the sense that they could be regarded as the Baer sum of the pull-back of the canonical cover of $\operatorname {SL}_{r+1}(F)$ and a determinantal cover. Moreover, there exists a functorial lift from representations of a KP-cover to that of G; see, for instance, [Reference Flicker and Kazhdan28], [Reference Zou80], etc.

• • (Savin’s cover) When $\boldsymbol {c}=-1$ and $\boldsymbol {d}=2$ , we get a special cover constructed by Gordan Savin (S-cover for short); see, for instance, [Reference F. Gao and Weissman30, §4.1]. It can be realized as follows: first, we construct the canonical cover of the symplectic group $\operatorname {Sp}_{2r}(F)$ with respect to the Steinberg symbol $(\cdot ,\cdot )_{n}^{-1}$ , and then after identifying G with the Siegel Levi subgroup of $\operatorname {Sp}_{2r}(F)$ , we get the S-cover of G. Every Levi subgroup of G is block compatible, which can be seen from formula (4.8) since $2\boldsymbol {c}+\boldsymbol {d}=0$ .

5.1 Lattice sequences and lattice chains

Fix an r-dimensional vector space V over F and a basis $\{v_{1},\dots ,v_{r}\}$ of V under which we identify G with $\operatorname {Aut}_{F}(V)$ . We write $A=\operatorname {End}_{F}(V)\cong \operatorname {M}_{r}(F)$ . For $\beta \in A$ , we define $\psi _{\beta }(x)=\psi _{F}(\operatorname {tr}(\beta (x-1)))$ for $x\in A$ .

An $\mathfrak {o}_{F}$ -lattice sequence $\Lambda =(\Lambda _{k})_{k\in \mathbb {Z}}$ of V is a sequence of $\mathfrak {o}_{F}$ -lattices of V, such that

• • $\Lambda _{k+1}\subset \Lambda _{k}$ for any $k\in \mathbb {Z}$ .

• • There exists a positive integer $e=e(\Lambda |\mathfrak {o}_{F})$ , called the period of $\Lambda $ over $\mathfrak {o}_{F}$ , such that $\Lambda _{k+e}=\mathfrak {p}_{F}\Lambda _{k}$ for any $k\in \mathbb {Z}$ .

When $\Lambda _{k+1}\subsetneq \Lambda _{k}$ for each $k\in \mathbb {Z}$ , we call $\Lambda $ a strict lattice sequence, or a lattice chain of V. We denote by $\mathscr {L}(V,\mathfrak {o}_{F})$ (resp. $\mathscr {L}^{+}(V,\mathfrak {o}_{F})$ ) the set of $\mathfrak {o}_{F}$ -lattice sequences (resp. $\mathfrak {o}_{F}$ -lattice chains) of V.

We may realize a lattice sequence $\Lambda $ as a function defined on $\mathbb {R}$ by imposing $\Lambda _{x}=\Lambda _{\lceil x\rceil }$ for $x\in \mathbb {R}$ .

For $\Lambda \in \mathscr {L}(V,\mathfrak {o}_{F})$ , we define a lattice sequence $\mathfrak {A}(\Lambda )\in \mathscr {L}(A,\mathfrak {o}_{F})$ by

$$ \begin{align*}\mathfrak{A}_{k}(\Lambda)=\{a\in A\mid a\Lambda_{l}\subset\Lambda_{l+k},\ l\in\mathbb{Z} \},\quad k\in\mathbb{Z}.\end{align*} $$

In particular, $\mathfrak {a}=\mathfrak {A}_{0}(\Lambda )$ is a hereditary order in A, and $\mathfrak {p}_{\mathfrak {a}}=\mathfrak {A}_{1}(\Lambda )$ is the Jacobson radical of $\mathfrak {a}$ . We define the valuation map

$$ \begin{align*}v_{\Lambda}:A\rightarrow \mathbb{Z},\ x\mapsto\mathrm{max}\{k\in\mathbb{Z}\mid x\in\mathfrak{A}_{k}(\Lambda)\}\end{align*} $$

with the convention $v_{\Lambda }(0)=\infty $ . We write $U(\Lambda )=\mathfrak {A}_{0}^{\times }$ and $U_{k}(\Lambda )=1+\mathfrak {A}_{k}(\Lambda )$ for $k\geq 1$ .

Up to the choice of an F-basis of V, we may realize $\mathfrak {a}$ as a standard hereditary order in $\operatorname {M}_{r}(F)$ , meaning that there exists a composition $r_{1}+\dots +r_{t}=r$ such that

$$ \begin{align*}\mathfrak{a}=\{(a_{ij})_{1\leq i,j\leq t}\mid a_{ij}\in\operatorname{M}_{r_{i}\times r_{j}}(\mathfrak{o}_{F})\ \text{for}\ 1\leq i\leq j\leq t\ \text{and}\ a_{ij}\in\operatorname{M}_{r_{i}\times r_{j}}(\mathfrak{p}_{F})\ \text{for}\ 1\leq j< i\leq t\}.\end{align*} $$

If $\Lambda $ is strict, then $\mathfrak {A}_{k}(\Lambda )=\mathfrak {p}_{\mathfrak {a}}^{k}$ for any $k\in \mathbb {Z}$ . Indeed, it is not hard to show that $\mathscr {L}^{+}(V,\mathfrak {o}_{F})\rightarrow \mathscr {L}^{+}(A,\mathfrak {o}_{F}),\ \Lambda \rightarrow \mathfrak {A}(\Lambda )$ is a bijection, so we may somehow recover the corresponding lattice chain from a given hereditary order $\mathfrak {a}$ in A.

Let E be a subfield of A over F. Then we may realize V as a vector space over E, which we denote by $V_{E}$ . Let $B=\operatorname {End}_{E}(V)$ , which is an F-subalgebra of A. Then $B^{\times }$ is the centralizer of $E^{\times }$ in G. Let $\psi _{E}$ be a character of E of conductor $\mathfrak {p}_{E}$ . Then there exists a bi-B-module homomorphism $s:A\rightarrow B$ such that

$$ \begin{align*}\psi_{E}(\operatorname{tr}_{E}(s(a)b))=\psi_{F}(\operatorname{tr}_{F}(ab)),\quad a\in A,\ b\in B,\end{align*} $$

called a tame corestriction on A.

A sequence $\Lambda \in \mathscr {L}(V,\mathfrak {o}_{F})$ is called E-pure if it is normalized by $E^{\times }$ . In this case, each $\Lambda _{k}$ is an $\mathfrak {o}_{E}$ -lattice. Moreover, $\Lambda $ could be realized as an element in $\mathscr {L}(V_{E},\mathfrak {o}_{E})$ , which we denote by $\Lambda _{E}$ . By definition, $\mathfrak {A}_{k}(\Lambda _{E})=\mathfrak {A}_{k}(\Lambda )\cap B$ for each $k\in \mathbb {Z}$ . Let $\mathfrak {b}=\mathfrak {A}_{0}(\Lambda _{E})$ , which is a hereditary order in B. Let $\mathfrak {p}_{\mathfrak {b}}=\mathfrak {A}_{1}(\Lambda _{E})$ .

We consider a decomposition $V=\bigoplus _{i=1}^{t} V^{i}$ of F-vector spaces. It is called an E-decomposition if each $V^{i}$ is E-stable. In this case, we have $V_{E}=\oplus _{i=1}^{t}V_{E}^{i}$ .

Let $M=\prod _{i=1}^{t}\operatorname {Aut}_{F}(V^{i})\subset G$ and let $P=MN$ be a parabolic subgroup of G having a Levi factor M and the unipotent radical N. For an E-decomposition $V=\bigoplus _{i=1}^{t} V^{i}$ , we get a parabolic subgroup $P_{E}=P\cap B^{\times }$ of $B^{\times }$ , having a Levi factor $M_{E}=M\cap B^{\times }$ and the unipotent radical $N_{E}=N\cap B^{\times }$ .

Let $\Lambda ^{i}\in \mathscr {L}(V^{i},\mathfrak {o}_{F})$ for $i=1,\dots t$ . Then the direct sum $\Lambda =\bigoplus _{i=1}^{t}\Lambda ^{i}$ is well-defined as an element in $\mathscr {L}(V,\mathfrak {o}_{F})$ . On the other hand, we say that a decomposition $V=\bigoplus _{i=1}^{t} V^{i}$ conforms with $\Lambda \in \mathscr {L}(V,\mathfrak {o}_{F})$ if there exists $\Lambda ^{i}\in \mathscr {L}(V^{i},\mathfrak {o}_{F})$ for each i such that $\Lambda =\bigoplus _{i=1}^{t}\Lambda ^{i}$ . In this case, we necessarily have $\Lambda ^{i}=\Lambda \cap V^{i}$ . Furthermore, if $V=\bigoplus _{i=1}^{t} V^{i}$ is an E-decomposition and each $\Lambda ^{i}$ is E-pure, we get the decomposition $\Lambda _{E}=\bigoplus _{i=1}^{t}\Lambda _{E}^{i}$ .

5.2 Strata

A stratum in A is a $4$ -tuple $[\Lambda ,u,l,\beta ]$ , where $\Lambda \in \mathscr {L}(V,\mathfrak {o}_{F})$ , and $u,l$ are integers such that $0\leq l\leq u-1$ , and $\beta \in \mathfrak {A}_{-u}(\Lambda )$ .

If the stratum we consider is strict, saying that $\Lambda $ is strict, then conventionally we often write $[\mathfrak {a},u,l,\beta ]$ for the same stratum to emphasize this assumption, where $\mathfrak {a}=\mathfrak {A}_{0}(\Lambda )$ . Conventionally, if an object $\Gamma (\Lambda ,\beta ,\dots )$ is defined for a stratum with $\Lambda $ strict, we often write $\Gamma (\mathfrak {a},\beta ,\dots )$ instead.

Two strata $[\Lambda ,u,l,\beta _{1}]$ and $[\Lambda ,u,l,\beta _{2}]$ are called equivalent if $\beta _{1}-\beta _{2}\in \mathfrak {A}_{-l}(\Lambda )$ .

For a stratum $[\Lambda ,u,l,\beta ]$ in A such that $0\leq l<u\leq 2l+1$ , we notice that $\psi _{\beta }$ becomes a character of $U^{l+1}(\Lambda )/U^{u+1}(\Lambda )$ , which depends only on the equivalence class of $[\Lambda ,u,l,\beta ]$ .

Assume that $[\Lambda ,u,l,\beta ]$ is pure, which means that $E:=F[\beta ]$ is a field and $\Lambda $ is E-pure. In this case, we let B be the centralizer of E in A. Once a related tame corestriction $s:A\rightarrow B$ is fixed, a derived stratum of $[\Lambda ,u,l,\beta ]$ is a stratum $[\Lambda _{E},l,l-1,s(c)]$ in B with $c\in \mathfrak {A}_{-l}(\Lambda )$ .

For $k\in \mathbb {Z}$ , we define

$$ \begin{align*}\mathfrak{n}_{k}(\beta,\Lambda)=\{x\in\mathfrak{a}\mid \beta x-x\beta\in \mathfrak{A}_{k}(\Lambda)\}.\end{align*} $$

If $E\neq F$ , there exists a maximal integer k such that $\mathfrak {n}_{k}(\beta ,\Lambda )$ is not contained in $\mathfrak {b}+\mathfrak {p}_{\mathfrak {a}}$ , which we denote by $k_{0}(\beta ,\Lambda )$ . Then $k_{0}(\beta ,\Lambda )\geq -u$ since $\mathfrak {n}_{-u}(\beta ,\Lambda )=\mathfrak {a}$ . If $E=F$ , we let $k_{0}(\beta ,\Lambda )=-\infty $ . Then we call a pure stratum $[\Lambda ,u,l,\beta ]$ simple if $l<-k_{0}(\beta ,\Lambda )$ .

Assume that we have an E-decomposition $V=\bigoplus _{i=1}^{t}V^{i}$ . Fix a simple stratum $[\Lambda ,u,l,\beta ]$ in A and assume that $\Lambda $ conforms with this decomposition. Then each $[\Lambda ^{i},u,l,\beta ]$ is a simple stratum in $A^{i}=\operatorname {End}_{F}(V^{i})$ for $i=1,\dots ,t$ (cf. [Reference Sécherre and Stevens66, Proposition 1.20]).

In particular, we consider a stratum of the form $[\Lambda ,u,u-1,b]$ . Let $e=e(\Lambda |\mathfrak {o}_{F})$ . The element $\varpi _{F}^{u/\operatorname {gcd}(u,e)}b^{e/\operatorname {gcd}(u,e)}$ is in $\mathfrak {a}^{\times }$ . So its characteristic polynomial has coefficients in $\mathfrak {o}_{F}$ , whose reduction module $\mathfrak {p}_{F}$ in $\boldsymbol {k}[X]$ is called the characteristic polynomial of $[\Lambda ,u,u-1,b]$ and denoted by $\varphi _{b}$ .

Then a stratum $[\Lambda ,u,u-1,b]$ is called

• • fundamental if $\varphi _{b}$ is not a power of X.

• • split if $\varphi _{b}$ has at least two different irreducible factors.

• • minimal if it is simple or, in other words, $k_{0}(b,\Lambda )=-u$ .

Finally, we also allow the occurrence of a ‘null’ stratum $[\Lambda ,0,0,\beta ]$ , where $\Lambda $ is a lattice chain such that the corresponding hereditary order $\mathfrak {a}$ is maximal in A, and $\beta \in \mathfrak {o}_{F}$ . In this case, we have $E=F$ and $A=B$ .

5.3 Simple characters

Let $[\Lambda ,u,0,\beta ]$ be a simple stratum in $A=\operatorname {End}_{F}(V)$ and let $E=F[\beta ]$ . We first assume $\Lambda $ , as well as the occurring lattice sequences, to be strict. As in [Reference Bushnell and Kutzko19, §3] and [Reference Sécherre and Stevens66, §2],

1. (1) We define two sub- $\mathfrak {o}_{F}$ -orders $\mathfrak {H}(\beta ,\Lambda )\subset \mathfrak {J}(\beta ,\Lambda )$ of $\mathfrak {a}$ , which depend only on the equivalence class of $[\Lambda ,u,0,\beta ]$ . Each of them is filtered by a sequence of bilateral ideals correspondingly:

   $$ \begin{align*}\mathfrak{H}^{k}(\beta,\Lambda)=\mathfrak{H}(\beta,\Lambda)\cap\mathfrak{A}_{k}(\Lambda),\quad \mathfrak{J}^{k}(\beta,\Lambda)=\mathfrak{J}(\beta,\Lambda)\cap\mathfrak{A}_{k}(\Lambda),\quad k\geq 1.\end{align*} $$

   We denote by $H(\beta ,\Lambda )$ (resp. $J(\beta ,\Lambda )$ ) the subgroup of invertible elements in $\mathfrak {H}(\beta ,\Lambda )$ (resp. $\mathfrak {J}(\beta ,\Lambda )$ ). Then similarly, each of them is filtered by a sequence of open compact subgroups

   $$ \begin{align*}H^{k}(\beta,\Lambda)=H(\beta,\Lambda)\cap U^{k}(\Lambda),\quad J^{k}(\beta,\Lambda)=J(\beta,\Lambda)\cap U^{k}(\Lambda),\quad k\geq 1.\end{align*} $$

   We also have $J(\beta ,\Lambda )=U(\Lambda _{E})J^{1}(\beta ,\Lambda )$ .

2. (2) For each $0\leq l\leq k_{0}(\beta ,\Lambda )-1$ , we define a finite set $\mathcal {C}(\Lambda ,l,\beta )$ of characters of $H^{l+1}(\beta ,\Lambda )$ , called simple characters of level l attached to $[\Lambda ,u,0,\beta ]$ . We write $\mathcal {C}(\Lambda ,\beta )=\mathcal {C}(\Lambda ,0,\beta )$ for short. We remark that $\mathcal {C}(\Lambda ,l,\beta )$ depends on the choice of $\psi _{F}$ .

3. (3) Let $v=-k_{0}(\beta ,\Lambda )>0$ . For $k\geq 1$ , we define

   (5.1) $$ \begin{align} \mathfrak{m}_{k}(\beta,\Lambda)=\mathfrak{A}_{k}(\Lambda)\cap\mathfrak{n}_{k-v}(\beta,\Lambda)+\mathfrak{J}^{\lceil v/2\rceil}(\beta,\Lambda)\quad\text{and}\quad\Omega_{k}(\beta,\Lambda)=1+\mathfrak{m}_{k}(\beta,\Lambda). \end{align} $$

   Then for $\theta \in \mathcal {C}(\Lambda ,l,\beta )$ , we have

   (5.2) $$ \begin{align} I_{G}(\theta)=\Omega_{v-l}(\beta,\Lambda)B^{\times}\Omega_{v-l}(\beta,\Lambda). \end{align} $$

   In particular, if $l=0$ , we have

   (5.3) $$ \begin{align} I_{G}(\theta)=J(\beta,\Lambda)B^{\times}J(\beta,\Lambda). \end{align} $$

4. (4) Let V, $V'$ be two finite dimensional F-vector spaces, let $A=\operatorname {End}_{F}(V)$ and $A'=\operatorname {End}_{F}(V')$ , let $E=F[\beta ]$ and $E'=F[\beta ']$ be subfields of A and $A'$ , respectively, with an isomorphism $E\rightarrow E'$ mapping $\beta $ to $\beta '$ . Let $[\Lambda ,u,0,\beta ]$ (resp. $[\Lambda ',u',0,\beta ']$ ) be a strict simple stratum in A (resp. $A'$ ). Then we have a transfer map as a bijection

   $$ \begin{align*}t_{\mathfrak{a},\mathfrak{a}',\beta,\beta'}:\mathcal{C}(\Lambda,l,\beta)\rightarrow\mathcal{C}(\Lambda',l',\beta'),\end{align*} $$

   where we assume $0\leq l\leq u-1$ , $0\leq l\leq u'-1$ and $\lfloor l/e(\Lambda |\mathfrak {o}_{E})\rfloor =\lfloor l'/e(\Lambda '|\mathfrak {o}_{E})\rfloor $ .

   In particular, in the following cases, this transfer map could be further characterized.

   1. (a) Assume $V=V'$ , $l=l'$ , and there exists $g\in \operatorname {Aut}_{F}(V)$ such that $\beta '=\beta ^{g}$ and $\mathfrak {a}'=\mathfrak {a}^{g}$ . Then the transfer map is given by the g-conjugation.

   2. (b) Assume $V=V'$ and $\beta =\beta '$ . Then for $\theta \in \mathcal {C}(\Lambda ,l,\beta )$ , the transfer of $\theta $ to $\mathcal {C}(\Lambda ',l',\beta )$ coincides with $\theta $ on $H^{l+1}(\beta ,\Lambda )\cap H^{l'+1}(\beta ,\Lambda ')$ .

   3. (c) Assume that $V=V'\oplus V"$ is an E-decomposition, and there exists an E-pure lattice chain $\Lambda "$ in $V"$ such that $\Lambda =\Lambda '\oplus \Lambda "$ . Then for any $l\leq u-1$ , we have $H^{l}(\beta ,\Lambda )\cap \operatorname {Aut}_{F}(V')=H^{l}(\beta ',\Lambda ')$ . Moreover, for $l'=l$ , the transfer map is given by the restriction to $H^{l}(\beta ',\Lambda ')$ .

5. (5) Let $V=\bigoplus _{i=1}^{t}V^{i}$ be an E-decomposition with $E=F[\beta ]$ . Let $\Lambda ^{i}$ be an E-pure lattice chainFootnote ³ in $V^{i}$ for each i, such that $\Lambda =\bigoplus _{i=1}^{t}\Lambda ^{i}$ . Then each $[\Lambda ^{i},u,0,\beta ]$ is a simple stratum in $A^{i}=\operatorname {End}_{F}(V^{i})$ . Let $M=\prod _{i=1}^{t}\operatorname {Aut}_{F}(V^{i})$ be the corresponding Levi subgroup of G, and let $P=MN$ be a parabolic subgroup of G having a Levi factor M. Then for $\theta \in \mathcal {C}(\Lambda ,l,\beta )$ ,

   1. (a) The pair $(H^{l+1}(\beta ,\Lambda ),\theta )$ is decomposed with respect to $(M,P')$ for any parabolic subgroup $P'$ of G with a Levi factor M (cf. §3.4).

   2. (b) Moreover, we have

      (5.4) $$ \begin{align} H^{l+1}(\beta,\Lambda)\cap M=\prod_{i=1}^{t}H^{l+1}(\beta,\Lambda^{i})\quad\text{and}\quad \theta_{M}:=\theta\lvert_{H^{l+1}(\beta,\Lambda)\cap M}=\theta_{1}\otimes\dots\otimes\theta_{t}, \end{align} $$
      with each $\theta _{i}$ being the transfer of $\theta $ to $\mathcal {C}(\Lambda ^{i},l,\beta )$ .

Now we consider a general lattice sequence $\Lambda $ , and we explain how to generalize the above theory in this case. We refer to [Reference Sécherre and Stevens66, §2] for the missing details.

We construct an E-decomposition of F-vector spaces $V"=V\oplus V'$ . Write $A"=\operatorname {End}_{F}(V")$ , which contains $A=\operatorname {End}_{F}(V)$ as a sub-F-algebra. For $e=e(\Lambda |\mathfrak {o}_{F})$ , we choose a strict E-pure lattice sequence $\Lambda '$ in $V'$ of period e. Then the E-pure lattice sequence $\Lambda "=\Lambda \oplus \Lambda '$ in $V"$ is also strict and of period e. Moreover, $[\Lambda ",u,0,\beta ]$ (after identifying $\beta $ with an element in $A"$ ) is a simple stratum in $A"$ .

We define

$$ \begin{align*}\mathfrak{H}(\beta,\Lambda)=\mathfrak{H}(\beta,\Lambda")\cap A\quad\text{and}\quad\mathfrak{J}(\beta,\Lambda)=\mathfrak{J}(\beta,\Lambda")\cap A\end{align*} $$

as $\mathfrak {o}_{F}$ -orders in A. Then, we define $\mathfrak {H}^{k}(\beta ,\Lambda )$ , $\mathfrak {J}^{k}(\beta ,\Lambda )$ , $H(\beta ,\Lambda )$ , $J(\beta ,\Lambda )$ , $H^{k}(\beta ,\Lambda )$ , $J^{k}(\beta ,\Lambda )$ , $\mathfrak {m}_{k}(\beta ,\Lambda )$ , $\Omega _{k}(\beta ,\Lambda )$ as above.

We have $H^{l+1}(\beta ,\Lambda ")\cap A=H^{l+1}(\beta ,\Lambda )$ for any $l\geq 0$ . Moreover, for $0\leq l<-k_{0}(\beta ,\Lambda ")$ , we define simple characters of level l attached to $[\Lambda ,u,0,\beta ]$ to be the restriction of characters in $\mathcal {C}(\Lambda ",l,\beta )$ to $H^{l+1}(\beta ,\Lambda )$ . We denote by $\mathcal {C}(\Lambda ,l,\beta )$ the set of such characters.

We remark that the above notation is well-defined (i.e., independent of the choice of $V'$ , $V"$ and $\Lambda '$ ), and it is compatible with the original theory when $\Lambda $ is strict. Moreover, all the results for simple characters listed above remain valid.

For any simple stratum $[\Lambda ,u,0,\beta ]$ in A, we may define a related strict simple stratum $[\mathfrak {a},u',0,\beta ]$ in A, where $\mathfrak {a}=\mathfrak {A}_{0}(\Lambda )$ and $u'=-v_{\mathfrak {a}}(\beta )$ .

We also briefly recall the concept ‘endo-class’ introduced by Bushnell and Henniart [Reference Bushnell and Henniart17], although it will not be essentially used in this article. Let $V_{1}$ , $V_{2}$ be two finite dimensional vector spaces over F. Let $[\mathfrak {a}_{1},u_{1},0,\beta _{1}]$ (resp. $[\mathfrak {a}_{2},u_{2},0,\beta _{2}]$ ) be a strict simple stratum in $A_{1}=\operatorname {End}_{F}(V_{1})$ (resp. $A_{2}=\operatorname {End}_{F}(V_{2})$ ). We call $\theta _{1}\in \mathcal {C}(\mathfrak {a}_{1},0,\beta _{1})$ and $\theta _{2}\in \mathcal {C}(\mathfrak {a}_{2},0,\beta _{2})$ endo-equivalent if there exist an F-vector space V containing $V_{1}$ and $V_{2}$ , and two simple strata $[\mathfrak {a}_{1}',u_{1},0,\beta _{1}']$ and $[\mathfrak {a}_{2}',u_{2},0,\beta _{2}']$ in $\operatorname {End}_{F}(V)$ , with $F[\beta _{1}]\cong F[\beta ']$ and $F[\beta _{2}]\cong F[\beta _{2}']$ mapping $\beta _{1}$ to $\beta _{1}'$ and $\beta _{2}$ to $\beta _{2}'$ , such that the transfers $t_{\mathfrak {a}_{1},\mathfrak {a}_{1}',\beta _{1},\beta _{1}'}(\theta _{1})$ and $t_{\mathfrak {a}_{2},\mathfrak {a}_{2}',\beta _{2},\beta _{2}'}(\theta _{2})$ are conjugate by $G=\operatorname {Aut}_{F}(V)$ .

We notice that such an equivalence relation is independent of various choices above. Thus, we introduced an equivalence relation on

$$ \begin{align*}\bigcup_{[\mathfrak{a},u,0,\beta]}\mathcal{C}(\mathfrak{a},0,\beta),\end{align*} $$

where the union ranges over all the strata in $\operatorname {M}_{r}(F)$ with r ranging over all positive integers. Such an equivalence class is called an endo-class.

Finally, for a ‘null’ stratum $[\Lambda ,0,0,\beta ]$ in A and $\beta \in \mathfrak {o}_{F}$ , by convention we have $\mathfrak {H}(\beta ,\mathfrak {a})=\mathfrak {J}(\beta ,\mathfrak {a})=\mathfrak {a}$ , $\mathfrak {H}^{i}(\beta ,\mathfrak {a})=\mathfrak {J}^{i}(\beta ,\mathfrak {a})=\mathfrak {p}_{\mathfrak {a}}^{i}$ , $H(\beta ,\mathfrak {a})=J(\beta ,\mathfrak {a})=U(\mathfrak {a})$ , $H^{i}(\beta ,\mathfrak {a})=J^{i}(\beta ,\mathfrak {a})=U^{i}(\mathfrak {a})$ and $\mathcal {C}(\mathfrak {a},\beta )=\{1\}$ . In this case, the corresponding endo-class is trivial.

5.4 Heisenberg representations and $\beta $ -extensions

The reference here is [Reference Bushnell and Kutzko19, §5.1, 5.2, 7.1, 7.2] and [Reference Sécherre64]. Let $[\mathfrak {a},u,0,\beta ]$ be a strict simple stratum in $A=\operatorname {End}_{F}(V)$ , where we denote by $\Lambda $ the lattice chain related to $\mathfrak {a}$ . Still we write $E=F[\beta ]$ and $B=\operatorname {End}_{E}(V_{E})$ . Let $\theta \in \mathcal {C}(\mathfrak {a},\beta )$ .

There exists a unique irreducible representation $\eta $ of $J^{1}(\beta ,\mathfrak {a})$ , called the Heisenberg representation of $\theta $ , whose restriction to $H^{1}(\beta ,\mathfrak {a})$ contains $\theta $ . The restriction of $\eta $ to $H^{1}(\beta ,\mathfrak {a})$ is isomorphic to $[J^{1}(\beta ,\mathfrak {a}):H^{1}(\beta ,\mathfrak {a})]^{1/2}\cdot \theta $ . Also, we have

(5.5) $$ \begin{align} I_{G}(\eta)=I_{G}(\theta)=J(\beta,\mathfrak{a})B^{\times}J(\beta,\mathfrak{a}). \end{align} $$

Moreover, each related intertwining space of $\eta $ is of dimension $1$ .

A representation $\kappa $ of $J(\beta ,\mathfrak {a})=U(\mathfrak {b})J^{1}(\beta ,\mathfrak {a})$ is called a $\beta $ -extension of $\eta $ if its restriction to $J^{1}(\beta ,\mathfrak {a})$ is $\eta $ , and moreover, $I_{G}(\kappa )=I_{G}(\eta )$ . We notice that such $\kappa $ exists, and all the other $\beta $ -extensions of $\eta $ are of the form $\kappa \chi $ , where $\chi $ ranges over characters of $J(\beta ,\mathfrak {a})/J^{1}(\beta ,\mathfrak {a})\cong U(\mathfrak {b})/U^{1}(\mathfrak {b})$ that factor through $\operatorname {det}_{E}:B^{\times }\rightarrow E^{\times }$ .

We define $\boldsymbol {J}(\beta ,\mathfrak {a})=N_{B^{\times }}(U(\mathfrak {b}))J(\beta ,\mathfrak {a})$ as a compact modulo center subgroup of G. In particular, if $\mathfrak {b}$ is a maximal hereditary order, then $\boldsymbol {J}(\beta ,\mathfrak {a})=E^{\times }J(\beta ,\mathfrak {a})$ . Notice that $\boldsymbol {J}(\beta ,\mathfrak {a})$ is indeed the normalizer of the simple character $\theta $ ; thus, it does not depend on the choice of the simple stratum $[\mathfrak {a},u,0,\beta ]$ , but only on $\theta $ .

Now we consider E-pure lattice chains $\Lambda '$ and $\Lambda "$ in A, such that the corresponding hereditary orders $\mathfrak {a}'$ and $\mathfrak {a}"$ satisfy $\mathfrak {a}'\subset \mathfrak {a}\subset \mathfrak {a}"$ . Let $u'=-v_{\mathfrak {a}'}(\beta )$ and $u"=-v_{\mathfrak {a}"}(\beta )$ be two positive integers. Then $[\mathfrak {a}',u',0,\beta ]$ and $[\mathfrak {a}",u",0,\beta ]$ are simple strata in A. Let $\theta '\in \mathcal {C}(\mathfrak {a}',\beta )$ and $\theta "\in \mathcal {C}(\mathfrak {a}",\beta )$ be the corresponding transfers of $\theta $ , and let $\eta '$ and $\eta "$ be the Heisenberg representations of $\theta '$ and $\theta "$ , respectively. Then, there exist unique $\beta $ -extensions $\kappa '$ of $\eta '$ and $\kappa "$ of $\eta "$ , such that

(5.6) $$ \begin{align} \begin{aligned} &\operatorname{ind}_{U(\mathfrak{b}')J^{1}(\beta,\mathfrak{a})}^{U(\mathfrak{b}')U^{1}(\mathfrak{a}')}(\kappa\lvert_{U(\mathfrak{b}')J^{1}(\beta,\mathfrak{a})})\cong\operatorname{ind}_{J(\beta,\mathfrak{a}')}^{U(\mathfrak{b}')U^{1}(\mathfrak{a}')}(\kappa')\\ \text{and}\quad&\operatorname{ind}_{U(\mathfrak{b})J^{1}(\beta,\mathfrak{a}")}^{U(\mathfrak{b})U^{1}(\mathfrak{a})}(\kappa"\lvert_{U(\mathfrak{b})J^{1}(\beta,\mathfrak{a}")})\cong\operatorname{ind}_{J(\beta,\mathfrak{a})}^{U(\mathfrak{b})U^{1}(\mathfrak{a})}(\kappa) \end{aligned} \end{align} $$

as irreducible representations (cf. [Reference Bushnell and Kutzko19, §5.2.14]). We call $\kappa '$ (resp. $\kappa "$ ) the $\beta $ -extension related to $\kappa $ .

Finally, we consider an E-decomposition $V=\bigoplus _{i=1}^{t}V^{i}$ which conforms with $\Lambda $ . Let $M=\prod _{i=1}^{t}\operatorname {Aut}_{F}(V^{i})$ be the corresponding Levi subgroup of G. We fix a parabolic subgroup $P=MN$ of G having a Levi factor M.

Write $\Lambda ^{i}=\Lambda \cap V^{i}$ , $\mathfrak {a}^{i}=\mathfrak {A}_{0}(\Lambda ^{i})$ and $\mathfrak {b}^{i}=\mathfrak {A}_{0}(\Lambda _E^{i})$ for $i=1,\dots ,t$ .

Let $\theta _{i}\in \mathcal {C}(\Lambda ^{i},\beta )$ be defined as in (5.4). Let $\eta _{i}$ be the Heisenberg representation of $\theta _{i}$ as a representation of $J^{1}(\beta ,\Lambda ^{i})$ , and let $\eta _{M}=\eta _{1}\boxtimes \dots \boxtimes \eta _{t}$ be an irreducible representation of

$$ \begin{align*}J^{1}_{M}(\beta,\mathfrak{a}):=J^{1}(\beta,\mathfrak{a})\cap M=J^{1}(\beta,\Lambda^{1})\times\dots\times J^{1}(\beta,\Lambda^{t}).\end{align*} $$

We define

(5.7) $$ \begin{align} J^{1}_{P}(\beta,\mathfrak{a})=H^{1}(\beta,\mathfrak{a})(J^{1}(\beta,\mathfrak{a})\cap P)=(H^{1}(\beta,\mathfrak{a})\cap N^{-})\cdot(J^{1}(\beta,\mathfrak{a})\cap M)\cdot(J^{1}(\beta,\mathfrak{a})\cap N) \end{align} $$

as a pro-p-subgroup of $J^{1}(\beta ,\mathfrak {a})$ , then there exists a unique irreducible representation $\eta _{P}$ of $J^{1}_{P}(\beta ,\mathfrak {a})$ , such that

• • $(J^{1}_{P}(\beta ,\mathfrak {a}),\eta _{P})$ is decomposed with respect to $(M,P')$ for any parabolic subgroup $P'$ of G with a Levi factor M.

• • $\eta _{P}\lvert _{J^{1}_{M}(\beta ,\mathfrak {a})}=\eta _{M}$ .

• • $\operatorname {ind}_{J^{1}_{P}(\beta ,\mathfrak {a})}^{J^{1}(\beta ,\mathfrak {a})}\eta _{P}=\eta $ .