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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).

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Appendix: Some complementary series of A-class

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?

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Tadić, M. On unitarizability and Arthur packets. manuscripta math. (2021). https://doi.org/10.1007/s00229-021-01330-6

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Mathematics Subject Classification

  • 22E50
  • Secondary 11F70
  • 11S37