On unitarizability and Arthur packets

Abstract

In this paper we begin to explore the relation between the question of unitarizability of classical p-adic groups, and Arthur packets, starting from Tadić (Mem Am Math Soc 2020, to appear).

Appendix: Some complementary series of A-class

Complementary series form a considerable part of unitary duals of reductive groups. Among them, the simplest ones are one-dimensional complementary series, which in the case of classical groups are of the form

$$\begin{aligned} \nu ^x\sigma \rtimes \pi , \quad 0< x<\beta , \end{aligned}$$

(8.1)

where $\sigma$ and $\pi$ are irreducible unitarizable representations of a general linear and a classical group, respectively, such that all representations $\nu ^x\sigma \rtimes \pi , 0\le x<\beta$, are irreducible, and such that $\nu ^\beta \sigma \rtimes \pi$ is reducible.

Observe that for parameterising the continuous family of complementary series (8.1), it is enough to know lower and upper bounds of the complementary series, i.e. $\sigma$ and $\pi$ (such that $\sigma \rtimes \pi$ is irreducible), and further, the first reducibility exponent $\beta$.

C. Mœglin mentioned to us that it is possible that some complementary series representations can be of A-class. We present below an example of this type. Below $\rho , \sigma$ and $\alpha$ are as in section 3.14.

Lemma 8.3

Let $\alpha \ge 1$, $x\ge 0$ and $\alpha -x\in {\mathbb {Z}}_{>0}$. Then $[x]\rtimes \sigma$ is in an A-packet (we already know that for $x=\alpha$, both irreducible subquotients are in A-packets).

Proof

If $x=0$, then we are in the tempered situation, and the claim obviously holds (the A-parameter is $\psi _\sigma \oplus E_{1,1}^\rho \oplus E_{1,1}^\rho$). Therefore, we suppose $x>0$, which implies $\alpha >1$.

First we show that $[\alpha -1]\rtimes \sigma$ is in an A-packet if $\alpha >1$. Denote by $(\psi _\sigma ',\epsilon _\sigma ')$ the parameter obtained from $(\psi _\sigma ,\epsilon _\sigma )$ deforming $E_{2\alpha -3,1}^\rho$ to $E_{2\alpha -1,1}^\rho$ (then ${\text {Jord}}_\rho (\psi _\sigma ')$ ends with $(1,2\alpha -3),(2\alpha -1,1)$). We have $\sigma =\pi (\psi _\sigma ',\epsilon _\sigma ')$.

Denote by $(\psi _1,\epsilon _1)$ the parameter obtained from $(\psi _\sigma ',\epsilon _\sigma ')$ by deforming $E_{2\alpha -1,1}^\rho$ to $E_{2\alpha +1,1}^\rho$ (now ${\text {Jord}}_\rho (\psi _1)$ ends with $(1,2\alpha -3),(2\alpha +1,1)$). Then $\pi (\psi _1,\epsilon _1)=\delta ([\alpha ];\sigma )$.

Let $(\psi _2,\epsilon _2)$ be obtained from $(\psi _1,\epsilon _1)$ by deforming $E_{1,2\alpha -3}^\rho$ to $E_{1,2\alpha -1}^\rho$ (now ${\text {Jord}}_\rho (\psi _1)$ ends with $(1,2\alpha -1),(2\alpha +1,1)$). Then $\pi (\psi _2,\epsilon _2)=L([\alpha -1];\delta ([\alpha ];\sigma ))$. Denote by $>_{\psi _2}$ the standard order on ${\text {Jord}}_\rho (\psi _2)$.

Let $\psi _3$ be the A-parameter obtained from $\psi _2$ by replacing $(1,2\alpha +1)$ with $(1,2\alpha -1)$ (now ${\text {Jord}}_\rho (\psi _3)$ ends with $(2\alpha -1,1),(1,2\alpha -1)$). Denote by $>_{\psi _3}$ on ${\text {Jord}}_\rho (\psi _3)$ standard order which satisfies

$$\begin{aligned} (2\alpha -1,1)>_{\psi _3}(1,2\alpha -1) \end{aligned}$$

($>_{\psi _3}$ is an admissible order, but not natural; $\psi _3$ is a multiplicity one parameter, but not discrete).

We denote by $\varphi :{\text {Jord}}_\rho (\psi _2)\rightarrow {\text {Jord}}_\rho (\psi _3)$ the standard bijection which preserves order. This implies that it carries

$$\begin{aligned} (2\alpha +1,1)\mapsto (2\alpha -1,1) \end{aligned}$$

(on the remaining elements it is the identity). Now ${\text {Jord}}(\psi _2)$ dominates ${\text {Jord}}(\psi _3)$ with respect to $>_{\psi _3}$. Here we need to consider the matrix $X_{(\rho ,A,B,\zeta _{a,b})}^{\gg }$ (defined in section 5 of [31]), which is in our case a $1\times 1$ matrix $X_{(\rho ,\alpha -1,0,1)}^{\gg }=[\alpha ].$ We get the elements of $\Pi _{\psi _3}$ from ${\text {Jord}}(\psi _2)$ applying ${\text {Jac}}_{\alpha }$ to each element of $\Pi _{\psi _2}$ (the result is always either an irreducible representation or 0). Observe that

$$\begin{aligned}&L([\alpha -1];\delta ([\alpha ];\sigma )) \hookrightarrow [-(\alpha -1)]\rtimes \delta ([\alpha ];\sigma ) \nonumber \\&\quad \hookrightarrow [-(\alpha -1)]\times [\alpha ]\rtimes \sigma \cong [\alpha ] \times [-(\alpha -1)]\rtimes \sigma \cong [\alpha ]\times [\alpha -1]\rtimes \sigma .\nonumber \\ \end{aligned}$$

(8.2)

Now Frobenius reciprocity implies that ${\text {Jac}}_{\alpha }(L([\alpha -1];\delta ([\alpha ];\sigma )))= [\alpha -1]\rtimes \sigma .$ Therefore, $[\alpha -1]\rtimes \sigma$ is in the A-packet of $\psi _{3}$.

In a similar way we show next that $[\alpha -2]\rtimes \sigma$ is in an A-packet if $\alpha >2$. Denote now by $(\psi _\sigma ',\epsilon _\sigma ')$ the parameter obtained from $(\psi _\sigma ,\epsilon _\sigma )$ by deforming $E_{2\alpha -5,1}^\rho$ to $E_{1,2\alpha -5}^\rho$ (then ${\text {Jord}}_\rho (\psi _\sigma ')$ ends with $(1,2\alpha -5),(2\alpha -3,1),(2\alpha -1,1)$). We have $\sigma =\pi (\psi _\sigma ',\epsilon _\sigma ')$.

Denote by $(\psi _1,\epsilon _1)$ the parameter obtained from $(\psi _\sigma ',\epsilon _\sigma ')$ by deforming $E_{2\alpha -1,1}^\rho$ to $E_{2\alpha +1,1}^\rho$ and then $E_{2\alpha -3,1}^\rho$ to $E_{2\alpha 11,1}^\rho$ (now ${\text {Jord}}_\rho (\psi _1)$ ends with $(1,2\alpha -5),(2\alpha -1,1),(2\alpha +1,1)$). We get directly that $\pi (\psi _1,\epsilon _1)=\delta _{{\text {s.p.}}}([\alpha -1],[\alpha ];\sigma )$.

Let $(\psi _2,\epsilon _2)$ be obtained from $(\psi _1,\epsilon _1)$ by deforming $E_{1,2\alpha -5}^\rho$ to $E_{1,2\alpha -3}^\rho$ (now ${\text {Jord}}_\rho (\psi _1)$ ends with $(1,2\alpha -3),(2\alpha -1,1),(2\alpha +1,1)$). Then $\pi (\psi _2,\epsilon _2)=L([\alpha -2];\delta _{{\text {s.p.}}}([\alpha -1],[\alpha ];\sigma ))$. Denote by $>_{\psi _2}$ the standard order on ${\text {Jord}}_\rho (\psi _2)$.

Denote by $\psi _3$ A-parameter obtained from $\psi _2$ by replacing $(2\alpha -1,1)$ by $(2\alpha -3,1)$ and then $(2\alpha +1,1)$ by $(2\alpha -1,1)$ (now ${\text {Jord}}_\rho (\psi _3)$ ends with $(2\alpha -3,1),(1,2\alpha -3),(2\alpha -1,1)$). Denote by $>_{\psi _3}$ the standard order on ${\text {Jord}}_\rho (\psi _3)$ which satisfies

$$\begin{aligned} (2\alpha -3,1)>_{\psi _3}(1,2\alpha -3). \end{aligned}$$

Let $\varphi :{\text {Jord}}_\rho (\psi _2)\rightarrow {\text {Jord}}_\rho (\psi _3)$ be the standard bijection which preserves order. This implies that it carries

$$\begin{aligned} (2\alpha -1,1)\mapsto (2\alpha -3,1), \quad (2\alpha +1,1)\mapsto (2\alpha -1,1), \end{aligned}$$

(on the remaining elements it is the identity). Now ${\text {Jord}}(\psi _2)$ dominates ${\text {Jord}}(\psi _3)$ with respect to $>_{\psi _3}$. Here we need to consider the matrices $X_{(\rho ,A,B,\zeta _{a,b})}^{\gg }$, which in our case are $1\times 1$ matrices $X_{(\rho ,\alpha -2,0,1)}^{\gg }=[\alpha -1]$ and $X_{(\rho ,\alpha -1,0,1)}^{\gg }=[\alpha ]$. We need to apply them in descending order to $L([\alpha -2];\delta _{{\text {s.p.}}}([\alpha -1],[\alpha ];\sigma ))$, i.e. we need to apply ${\text {Jac}}_\alpha \circ {\text {Jac}}_{\alpha -1}$ to the last representation (and we will get either 0 or an element of $\Pi _{\psi _3}$).

Now we will compute the action of the above operator on $L([\alpha -2];\delta _{{\text {s.p.}}}([\alpha -1],[\alpha ];\sigma )$. Observe that

$$\begin{aligned}&L([\alpha -2];\delta _{{\text {s.p.}}}([\alpha -1],[\alpha ];\sigma ) \hookrightarrow [-(\alpha -2)]\rtimes \delta _{{\text {s.p.}}}([\alpha -1],[\alpha ];\sigma ) \nonumber \\&\quad \hookrightarrow [-(\alpha -2)]\times [\alpha -1]\rtimes \delta ([\alpha ];\sigma ) \cong [\alpha -1]\times [-(\alpha -2)]\rtimes \delta ([\alpha ];\sigma ).\nonumber \\ \end{aligned}$$

(8.3)

Note that $[-(\alpha -2)]\rtimes \delta ([\alpha ];\sigma )$ is irreducible. Now Frobenius reciprocity implies that

$$\begin{aligned} {\text {Jac}}_{\alpha -1}(L([\alpha -2];\delta _{{\text {s.p.}}}([\alpha -1],[\alpha ];\sigma ))) =[-(\alpha -2)]\rtimes \delta ([\alpha ];\sigma ). \end{aligned}$$

Further

$$\begin{aligned}&[-(\alpha -2)]\rtimes \delta ([\alpha ];\sigma ) \hookrightarrow [-(\alpha -2)]\times [\alpha ]\times \sigma \cong [\alpha ]\times [-(\alpha -2)]\times \sigma \\&\quad \cong [\alpha ]\times [\alpha -2]\times \sigma , \end{aligned}$$

and one directly concludes that ${\text {Jac}}_{\alpha }([-(\alpha -2)]\rtimes \delta ([\alpha ];\sigma ))=[\alpha -2]\times \sigma$. Therefore, $[\alpha -2]\rtimes \sigma$ is in the A-packet of $\psi _{3}$.

Continuing this procedure, we complete the proof of the lemma. $\square$

Definition 8.4

Suppose that an irreducible representation $\pi$ of a classical group $S_n$ is in an A-packet, and that there do not exist Speh representations $\tau _1,\dots ,\tau _k$ and an irreducible representation $\pi _0$ of a classical group $S_m$ with $m<n$, contained in some A-packet, such that

$$\begin{aligned} \pi \hookrightarrow \tau _1\times \dots \times \tau _k\rtimes \pi _0. \end{aligned}$$

Then $\pi _0$ will be called a primitive representation of A-type.

We will very briefly recall the notion of automorphic dual, which L. Clozel introduced in [6] (one can find more details and further references in the original Clozel paper).

We first recall of notion of the support of a unitary representation $\Pi$ of a locally compact group $\mathbf{G}$. An irreducible unitary representation $\pi$ of $\mathbf{G}$ is weakly contained in $\Pi$ if each diagonal matrix coefficient of $\pi$ on each compact subset of $\mathbf{G}$ can be approximated by finite sums of diagonal matrix coefficients of $\Pi$ (i.e. each diagonal matrix coefficient of $\pi$ on each compact subset of $\mathbf{G}$ is the limit of sums of diagonal matrix coefficients of $\Pi$). The support of $\Pi$ is the set of equivalence classes of all irreducible unitary representations $\pi$ of $\mathbf{G}$ which are weakly contained in $\Pi$.

Let G be a reductive group defined over an algebraic number field k (or more generally, over a global field k). Fix any place v of k and denote by $k_v$ the completion of k at v. The automorphic dual ${\widehat{G}}_{v,aut}$ is the support of the representations of the group $G(k_v)$ of $k_v$-rational points of G in the space of square integrable automorphic forms $L^2(G(k)\backslash H({\mathbb {A}}_k))$ (by right translations). We denote $F = k_v$.

Motivated by [27], we ask the following

Question 8.5. Is each primitive representation of A-type isolated in the automorphic dual?