CHARACTER FORMULA FOR THE SUPERCUSPIDAL REPRESENTATIONS OF GL$_l$ (Automorphic forms, automorphic representations and automorphic $L$-functions over algebraic groups)

大阪府立大総合科学部 高橋哲也 (TETSUYA TAKAHASHI) $U_{1}^{*}$ .When $E/F$ is non-Galois, we use the base change lift.Let $L/F$ be an unramified extension of degree $l-1$ .In $L$ , there exists a l-th root of unity and $EL/L$ is Galois.Therefore we can use the tools in Galois case for $\mathrm{G}\mathrm{L}_{l}(L)$ .Let $\mathrm{G}\mathrm{a}1(L/F)=\langle\tau\rangle$ .By the result of Bushnell-Henniart [3], there is a base change lift $\eta_{L}$ of $\eta_{\theta}$ to $H_{L}^{1}$ such that the twisted trace of $\eta_{L}$ by $\tau$ gives the trace of $\eta_{\theta}$ .(See Proposition 3.7 and Lemma 3.8).We remark that we need not assume the characteristic of $F$ is $0$ since we do not use the Arthur-Clozel base change lift [1].The method to calculate the twisted trace of $\eta_{L}$ is similar to that of Galois case.The complete character formula is stated as Theorem 3.12.Closing this introduction, we compare our formula with the known results.The same type of character formula for the division algebra case was given by Corwin, Moy and Sally, $\mathrm{J}\mathrm{r}\dot{\mathrm{i}}\mathrm{n}[6]$ and for $\mathrm{G}\mathrm{L}_{l}$ case by Debacker in [7].Their formulas agree with the result given in section 2. It contains some root numbers associated with a quadratic form.In this paper, we have determined it completely in section 3. Moreover we find the Kloosterman sum appears in the character formula.These are new results of this paper.In [22], the author gave the character formula of $\pi_{\theta}$ for $\mathrm{G}\mathrm{L}_{3}$ by using the decomposition of $\pi_{\theta}$ as $E^{\cross}$ -module.But this need the explicit matrix form of an inverse matrix which is hard to treat for large $l$ .We can simplify the proof of the main theorem, although we treat a general prime $l$ .Notation Let $F$ be a non-archimedean local field.We denote by $\mathcal{O}_{F},$ $P_{F},$ $\varpi_{F},$ $k_{F}$ and $v_{F}$ the maximal order of $F$ , the maximal ideal of $\mathcal{O}_{F}$ , a prime element of $P_{F}$ , the residue field of $F$ and the valuation of $F$ normalized by $v_{F}(\varpi_{F})=1$ .We set $q$ to be the number of elements in $k_{F}$ .Hereafter we fix an additive character $\psi$ of $F$ whose conductor is $P_{F}$ , i.e., $\psi$ is trivial on $P_{F}$ and not trivial on $\mathcal{O}_{F}$ .For an extension $E$ over $F$ , we denote by $\mathrm{t}\mathrm{r}_{E}$ , $n_{E}$ the trace and norm to $F$ respectively.We set $\psi_{E}=\psi\circ \mathrm{t}\mathrm{r}_{E}$ .The trace of matrix is denoted by Tr.For an irreducible admissible representation $\pi$ of $\mathrm{G}\mathrm{L}_{l}(F)$ , the conductoral exponent of $\pi$ is defined to be the integer $f(\pi)$ such that the local constant $\in(s, \pi, \psi)$ of is the form $aq^{-}s(f(\pi)-\downarrow)$ .