% Run under latex2e \documentstyle[12pt,euscript]{amsart} \def\cal#1{{\EuScript#1}} \textheight=574pt \textwidth=432pt \oddsidemargin=18.88pt \evensidemargin=18.88pt \topmargin=14.21pt \newtheorem{defi}{Definition} \newtheorem{lem}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{th}{Theorem} \newtheorem{cor}{Corollary} \newtheorem{rem}{Remark} \newtheorem{conj}{Conjecture} \newcommand{\rep}{representation} \newcommand{\reps}{representations} \newcommand{\Rep}{Representation} \newcommand{\2}{\frac{1}{2}} \newcommand{\B}{{\bf B}} \newcommand{\z}{\zeta} \newcommand{\w}{\omega} \newcommand{\s}{\langle} \def\GL{\mathrm{GL}} \def\Ind{\mathop{\mathrm{Ind}}\nolimits} \def\Frob{\mathop{\mathrm{Frob}}\nolimits} \def\supp{\mathop{\mathrm{supp}}\nolimits} \def\trace{\mathop{\mathrm{trace}}\nolimits} \def\End{\mathop{\mathrm{End}}\nolimits} \def\endproof{\unskip\nobreak\hfil\penalty50\hskip.5em\hbox{}\nobreak\hfill$\Box$} %reinstate \matrix from standard latex, rather than from amsart {\makeatletter \gdef\matrix#1{\null\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}} \begin{document} \title[Harmonic Analysis on the Twisted Finite Poincar\'e Half-Plane] {Harmonic Analysis on the Twisted Finite Poincar\'e Upper Half-Plane} \author{Jorge Soto-Andrade} \address{Jorge Soto-Andrade\\ Departamento de Matem\'aticas\\ Facultad de Ciencias\\ Universidad de Chile\\ Casilla 653\\ Santiago, Chile} \email{sotoandr@@mate.uncor.edu, sotoandr@@abello.dic.uchile.cl} \author{Jorge Vargas} \address{Jorge Vargas\\ FAMAF\\ Universidad Nacional de C\'ordoba\\ Ciudad Universitaria\\ C\'ordoba, Argentina} \email{vargas@@mate.uncor.edu} \thanks{Soto-Andrade was partially supported by FONDECYT Grants 92-1041 and 1940590, DTI -- U. Chile, ICTP, ECOS-France and NSF Grant DMS-9022140 at MSRI. \ Vargas was partially supported by CONICET, CONICOR, SecytUNC, ICTP and TWAS} \begin{abstract} We prove that the induced representation from a non trivial character of the Coxeter torus of $\GL (2,F) $, for a finite field $F$, is multiplicity-free; we give an explicit description of the corresponding (twisted) spherical functions and a version of the Heisenberg Uncertainty Principle. %for this space. \end{abstract} \maketitle \section{Introduction} Let $F$ be a finite field, with $q$ elements, and $E $ be its unique quadratic extension. Put $G = \GL (2,F)$ and denote by $K$ the Coxeter torus of $G$, realized as the subgroup of all matrices $m_z (z \in E^\times) $ of the maps $ w \mapsto zw (w \in E) $ with respect to a fixed $F$ - basis of $ E$. Recall that the finite homogeneous space $ {\cal H} = G/K$ may be looked upon as the finite analogue of (the double cover of) the classical Poincar\'e Upper Half Plane (see \cite{arca}). Harmonic analysis on $ \cal H$ amounts to decompose the induced representation $\Ind _K^G {\bf1} $ from the unit character {\bf1} of $K$ to $G$. We are interested here in the ``twisted'' version of this, i.e., the decomposition of the induced representation $\Ind ^G_K \Phi$ from a non (necessarily) trivial character $\Phi$ of $K$ to $G$. The real analogue of this case has been considered in \cite{eja}. We prove that this representation is multiplicity-free, taking advantage of the fact that this is so for $\Ind _K^G {\bf1}$ (see \cite{arca}) and reducing the computation of the multiplicities in $\Ind ^G_K \Phi $ to the ones in $\Ind _K^G {\bf1}$. We also give an explicit description of the corresponding (twisted) spherical functions. Finally, we give a version of the Heisenberg Uncertainty Principle \section{The Multiplicity One Theorem for $\Ind ^G_K \Phi$.} \subsection {The case $\Phi = \bf 1$} We consider first the special case $\Phi = \bf 1 $ in which the multiplicity one theorem follows from a geometric argument. In fact, we have $$ \Ind ^G_K \bf 1 \simeq (L^2( \cal H), \tau), $$ where $L^2(\cal H)) $ stands for the space of all complex functions on $\cal H$ endowed with the usual canonical scalar product, and $\tau$ denotes the natural representation of $G$ in $L^2(\cal H) $, defined by $(\tau_gf)(z) = f(g^{-1}.z) $, where $ z \mapsto g.z $ is the homographic action of $G$ on $\cal H$, given by $$ g.z = \frac {az + b} {cz + d } \hspace{3cm} \mbox{for} g = \left( \matrix{ a &b\cr c &d \cr}\right) \in G, z \in \cal H . $$ \begin{defi} For all $z,w \in \cal H $, we put $$ D(z,w) = \frac{N(z - w)}{N(z - \bar w)} $$ with the convention that $D(z,w) = \infty $ if $w = \bar z$. \end{defi} \begin{prop} $D$ is an orbit classifying invariant function for the homographic action of $G$ in $\cal H \times \cal H$. \end{prop} \begin{cor} The commuting algebra of $ (L^2( \cal H), \tau)$ is commutative. \end{cor} This follows from the fact that, the classifying invariant $D$ being symmetric, the $G$-orbits in $\cal H \times \cal H$ are also symmetric. \endproof \subsection{The case of general $\Phi$} Let's denote by $\phi$ the restriction of $\Phi$ to $F^\times$. We will prove that every twisting of an irreducible representation $ \pi^d_{\theta} $ of $G$ (where the superscript $d$ denotes the dimension of $\pi$ and $\theta$ its character parameter) by the character $(\Phi +\Phi^q) $ is isomorphic to a representation of the form $\pi^{d'}_{\theta'}+\pi^{d''}_{\theta''}$, when restricted to $K$. In fact we will work with the characters $\chi^d_\theta$ of the irreducible representations $\pi ^d_{\theta} $ of $G$, for which we keep the notations of \cite{tesis} or \cite{anna}. \begin{lem} %$\phi:=\Phi_{\vert F}$ On $K$ we have\\ $(\Phi +\Phi^q) \chi^{q}_{\alpha, \alpha} = \chi^{q-1}_{\Phi (\alpha \circ N)} + \chi^{q+1}_{\phi \alpha, \alpha}$,\\ $(\Phi +\Phi^q) \chi^{1}_{\alpha, \alpha} = \chi^{q+1}_{\phi \alpha, \alpha } - \chi^{q-1}_{\Phi (\alpha \circ N)}$,\\ $(\Phi +\Phi^q) \chi^{q+1}_{\alpha, \beta} = \chi^{q+1}_{\phi \alpha,\beta} + \chi^{q+1}_{\alpha, \phi \beta}$,\\ $(\Phi +\Phi^q) \chi^{q-1}_{\Lambda} = \chi^{q-1}_{\Phi \Lambda} +\chi^{q-1}_{\Phi^q \Lambda}.$ \endproof \end{lem} Now for a character $\chi$ of $G$, we have $ \chi \circ \Frob = \chi$ on $K$, as it follows from the character table. Therefore $ \sum_K \bar \Phi (k^q) \chi (k) = \sum_K \bar \Phi (k) \chi (k) $ because $\Frob $ is an involutive automorphism. Hence, the multiplicity of $\pi$ in $\Ind _K^G \Phi $ equals $ \2 \sum_K (\bar \Phi+\bar \Phi ^q) (k) \pi(k) $ and so it is just the average of the multiplicities in $\Ind ^G_K {\bf 1} $ of two representations of $G$ (one of which may be virtual!) \begin{rem} Put $\pi^{q+1}_{\alpha, \alpha} = \pi^q_{\alpha} + \pi^1_{\alpha}$ and $ \pi^{q-1}_{ \alpha \circ N} = \pi^q_{\alpha} -\pi^1_{\alpha}$ for every $\alpha \in (F^\times)^{\wedge}$. %\begin{rem} It is easy to check than in the degenerate cases $ \alpha = \beta$ (for $\pi = \pi^{q+1}_{\alpha, \beta}$ ) and $\Lambda = \Lambda^q$ (for $\pi = \pi^{q-1}_\Lambda$) we find for the multiplicities $m_1 (\pi) $ \begin{equation} m_1(\pi^{q+1}_{\alpha, \alpha}) = 1 \hspace{4cm} (\alpha \in (F^\times)^{\wedge}) \label{eq:mdegsp} \end{equation} and \begin{equation} m_1(\pi^{q-1}_{ \alpha \circ N}) = - \delta_{\alpha, 1} \hspace{4cm}(\alpha \in (F^\times)^{\wedge} ) \label{eq:mdegsd} \end{equation} \end{rem} Using the fact that the multiplicities of the irreducible representations of $G$ in $\Ind _K^G \bf 1$ are at most one and also equations (\ref{eq:mdegsp}) and (\ref{eq:mdegsd}), we get that the multiplicities are also at most one in the more general case of $\Ind _K^G \Phi $. \endproof \subsection{The multiplicities $ m_{\Theta,d}(\Phi)$ of $\pi^d_\Theta $ in $\Ind _K^G \Phi $ for general $\Phi \in (E^{\times})^{\wedge} $} In Table~1 below, $ \pi^d_\Theta$ denotes an irreducible representation of $G$, of dimension $d$ and parameter $\Theta$. Then $d \in \{ 1, q, q+1, q-1\}$ and $\Theta$ is of the form $\{\alpha, \beta \} $, with $ \alpha, \beta \in (F^{\times})^{\wedge}$ or $\{\Lambda, \Lambda^q \} $ with $ \Lambda \in (E^{\times})^{\wedge}$ \begin{table}[hb] \caption{The multiplicities $ m_{\Theta,d}(\Phi)$} \begin{center} \begin{tabular}{|l|l|} \hline $\pi^d_\Theta $ & $ m_{ \Theta, d }(\Phi)$ \\ \hline $\pi^1_{\alpha, \alpha} $ & $ \delta_{\alpha^2, \phi} $ \\ \hline $\pi^q_{\alpha, \alpha} $ & $\delta_{\alpha^2,\phi} -\delta_{\alpha \circ N,\Phi} $ \\ \hline $\pi^{q+1}_{\alpha, \beta} $ & $\delta_{\alpha\beta,\phi} $ \\ \hline $\pi^{q - 1}_{ \Lambda, \Lambda^q} $ & $\delta_{\lambda,\phi} - \delta_{\Lambda,\Phi} - \delta_{\Lambda^q,\Phi} $ \\ \hline \end{tabular} \end{center} \end{table} \noindent {\sc NOTATIONS.} Here $ \alpha, \beta \in (F^{\times})^{\wedge}$ with $ \alpha \neq \beta$ and $ \Phi, \Lambda \in (E^{\times})^{\wedge}$ with $ \Lambda \neq \Lambda^q $, and $\lambda$ (resp. $\phi$ ) denotes the restriction of the character $ \Lambda $ (resp. $ \Phi$ ) to $(F^{\times})^{\wedge}$. \section{The twisted spherical functions} \subsection{The averaging construction} In this section $G$ denotes an arbitrary finite group, $K$ a subgroup of $G$and $\Phi$ a one dimensional representation of $K$. We notice that the spherical functions for the representation $\Ind _K^G \Phi $ are obtained as weighted averages of the characters of $G$. More precisely: \begin{defi} Let $ L^1(G)$ be the group algebra of $G$, realized as the convolution algebra of all complex functions of $G$ and let $ L^1_\Phi(G,K)$ be the convolution algebra of all complex functions $f$ on $G$ such that $$ f(kgk') = \Phi(k)f(g)\Phi(k') $$ for all $g \in G, k,k' \in K $. For any $f \in L^1(G)$ put $$ (P_ \Phi f)(g) = \frac{1}{|K|} \sum_{k \in K} \Phi^{-1}(k)f(kg) $$ for all $g \in G$. \end{defi} Notice that the operator $P_\Phi$ is just convolution with the idempotent function $\varepsilon^\Phi_K \in L^1G $ which coincides with $|K|^{-1}\Phi$ on $K$ and vanishes elsewhere. Moreover $ L^1_\Phi(G,K) $ may be writen as $ \varepsilon^\Phi_K \ast L^1G \ast \varepsilon^\Phi_K $ and its elements $f$ are characterized by the properties $$ \varepsilon^\Phi_K \ast f = f = f \ast \varepsilon^\Phi_K $$ \begin{lem} Let $\chi$ be the character of an irreducible representation $\pi$ of $G$. Then $P_{\Phi} (\chi) (e) \neq 0$ iff $\pi$ appears in $\Ind _K^G \Phi$. \endproof \end{lem} \begin{lem} $P_{\Phi} (\chi) $ is a non-zero function iff it doesn't vanish for $g=e$. \endproof \end{lem} \begin{prop} The mapping $P_ \Phi $ is an algebra epimorphism from the center $Z(L^1G)) $ of the convolution algebra $L^1G $ onto the center $Z( L^1_\Phi(G,K))$ of the convolution algebra $ L^1_\Phi(G,K)$. \end{prop} {\em Proof:} We have $$ \begin{array}{lllll} P_ \Phi (f_1 \ast f_2) & = & \varepsilon^\Phi_K \ast (f_1 \ast f_2) & = &( f_1 \ast \varepsilon^\Phi_K )\ast f_2 \\ & = & (f_1 \ast \varepsilon^\Phi_K \ast \varepsilon^\Phi_K )\ast f_2 & = & ( \varepsilon^\Phi_K \ast f_1 )\ast ( \varepsilon^\Phi_K \ast f_2 ) \\ & = & P_ \Phi f_1 \ast P_ \Phi f_2. \end{array} $$ since $f_1$ is central and $\varepsilon^\Phi_K $ is idempotent. Moreover the dimension $d$ of the image of $ Z(L^1G))$ under $P_ \Phi $ is the number of irreducible characters $\chi$ of $G$ such that $P_\Phi(\chi) \neq 0$; but $P_\Phi (\chi) \neq 0$ iff $(P_\Phi \chi)(e) \neq 0$ and, the number $(P_\Phi (\chi))(e)$ being the multiplicity in $\Ind _K^G \Phi$ of the representation $\pi$ of $G$ whose character is $\chi$, we see that $d$ is just the number of irreducible representations $\pi$ of $G$ appearing in $\Ind _K^G \Phi$, i. e. the dimension of the center of $ L^1_\Phi(G,K)$. \endproof \begin{cor} The nonzero functions that satisfy the functional equation $$ h(x)h(y) = \int_K \bar {\Phi }(k) h(xky) \,dk $$ linearly span the center of the algebra $L^1_{\Phi} (G,K) $. \end{cor} {\em Proof:} The functions $h$ that satisfy the above functional equation are exactly the complex multiples of the functions $P_{\Phi} (\chi) $; for a proof (see \cite{4}). Therefore the corollary follows. \endproof \subsection{Explicit formulae for the twisted spherical functions} Define $$ S^\Phi_\Lambda (a) = - (q^2 - 1)^{-1} \sum_{(z,w) \in \Gamma_a}\Phi^{-1}(z)\Lambda(w) $$ for $ \Lambda \in (E^\times)^{\wedge} $ and $a \in F^\times$, where $\Gamma_a $ denotes the set of all $(z,w) \in E ^{\times} \times E ^{\times} $ such that $ N(w) = aN(z) $ and $Tr(w) = 2(a + 1)^{-1} Tr(z) $. Then the spherical function $ \zeta ^ \Phi_ \Lambda $ of $G$ associated to the cuspidal character $\chi^{q-1}_\Lambda $ of $G$ is given on the representatives $d(a,1) = \left( \matrix{ a & 0 \cr 0 & 1 \cr} \right) $ \hspace{0.5cm} $(a \in F^\times) $ for the $K$ - double cosets in $G$, by \[ \zeta ^ \Phi_ \Lambda (d(a,1)) = S^\Phi_\Lambda (a) + q(q + 1)^{-1} \delta_{a,1} \delta_{\lambda, \phi}, \] where $\lambda$ (resp. $\phi$) denotes the restriction of the character $\Lambda$ (resp. $ \Phi $) of $E^ \times$ to $ F^ \times$. Notice that $a = 1$ corresponds to the origin in $\cal H$ and $a = -1$ corresponds to the antipode of the origin in $\cal H $. %compute values for a=1 and a = -1 % (last one corresponds to the antipode of the origin) It is not difficult to check that these formulae for the spherical functions are equivalent to the ones given in \cite{arca} for the case $\Phi = 1$. %Achtung! Definition of $S^\Phi_\Lambda (a)$ needs a delta term for %a = 1, otherwise it %won't fit the right multiplicity, to wit %$S^\Phi_\Lambda (a)= \subsection{A new form for the cuspidal spherical functions for $\Phi = {\bf1 }$ ({\bf char $F$} $ \neq 2) $} For $a \neq 1 $, one has the following new expression for the spherical functions estimated in \cite{katz} $$\zeta^\Phi_\Lambda (a) = (q + 1)^{-1} \sum_{u \in U} \varepsilon (Tr(u) - (a + a^{-1}))( \varepsilon \omega)(u),$$ for $a \neq 1$, where $\varepsilon$ denotes the sign character of $ F^ \times$. \section{Heisenberg Uncertainty Principle} For this section, $G$ denotes an arbitrary finite group, $K$ any subgroup of $G$ and $\Phi $ any linear character of $K$. Let $ \hat G^{\Phi}$ be set of allthe equivalence classes of irreducible representations of $G$ that contain the character $\Phi$ when restricted to $K$. For each equivalence class we choose, once and for all, a representative $(\pi,V_{\pi})$. As usual, for each $f$ in $L^1(G)$, the Fourier Transform ${\cal F} (f) $, valued in the class $(\pi,V_{\pi})$, is the linear operator $\cal F$ in $V_{\pi}$ defined by $$ {\cal F} (f) (\pi):=\pi (f):=\frac{1}{|G|} \int_G f(g) \pi (g^{-1}) dg:= \frac{1}{|G|}\sum_{ g \in G} f(g) \pi(g^{-1}). $$ We recall the statement of the Plancherel theorem for a function $ f \,\in L^1_{\Phi}(G,K) $ $$ f(g) =\frac{1}{G} \sum_{ \pi \in \hat G^{\Phi}} d_{\pi} \trace(\pi(f) \pi (g)) ; $$ here $ g\, \in \, G $ arbitrary and $d_{\pi}:=\dim V_{\pi} $. For any complex valued function $f$ on $G$, let $ |\supp (f)| $ denote the number of elements of the support of $f$. That is, the number of points of $G$ where $f$ takes nonzero values. \begin{prop} [\bf Heisenberg Uncertainty Principle] For any nonzero function $f \,\, \in \,\, L^1_{\Phi} (G,K) $ we have $$ |\supp (f) |\,( \sum_{\pi \in \supp ({\cal F} (f) )} d_{\pi})\, \geq\, |G|. $$ Here $\supp (\cal F (f) ) $ is the subset of $\hat G^{\Phi} $ where $\cal F (f)$ does not vanish. \end{prop} {\em Proof:} For any function $f$ on $G$ we recall that $$ \Vert f \Vert^2_{2} = \sum_{x \in G} \vert f(x) \vert ^2 ;\,\, \Vert f \Vert_{\infty} = \max_{ x \in G} \vert f(x) \vert ;\,\, \Vert f \Vert_2^2 \leq \Vert f \Vert^2_{\infty} \vert \supp (f) \vert \eqno (*) $$ From now on, we fix a $G-$invariant inner product on $V_{\pi}$. Then $T^*$ denotes the adjoint of a linear operator $T$ on $V_{\pi}$ with respect to this inner product. Also $\Vert T \Vert $ denotes the Hilbert-Schmidt norm on $\End V_{\pi} $ defined by $\trace(TS^*),$ for $ S, T \in \End V_{\pi} $ Since $ f \in L^1_{\Phi}(G,K) $, as we pointed out before, the Plancherel Theorem says that we have that $ \supp (f) $ is contained in $ \hat G^{\Phi} $ and that $$ f(x) =\frac{1}{G} \sum_{ \pi \in \hat G^{\Phi}} d_{\pi} \trace(\pi(f) \pi (x)). $$ The Cauchy--Schwarz inequality applied to the Hilbert-Schmidt inner product says that the first of the two following inequallities is true, $$ \trace (\pi (f) \pi (x) ) \leq \Vert \pi (f) \Vert \Vert \pi (x) \Vert \leq \Vert \pi (f) \Vert, $$ the second inequality follows from the fact that $ \Vert T \Vert =1 $ for a unitary operator. Putting together the last two statements we get $$ \Vert f \Vert_{\infty} \leq \frac{1}{G} \sum_{\pi \in \hat G^{\Phi}} d_{\pi} \Vert {\cal F} (f)(\pi) \Vert $$ The classical Cauchy--Schwarz inequality and the fact that $ d_{\pi} = d_{\pi}^{\frac 12}d_{\pi}^{\frac 12 }$ imply that $$ \Vert f \Vert^2_{\infty} \leq \frac{1}{\vert G \vert^2} \sum_{\pi \in \hat G^{\Phi} } d_{\pi} \Vert {\cal F} (f)(\pi) \Vert^2 \,\, \sum_{\pi \in \supp({\cal F}(f))} d_{\pi}. $$ Now the $L^2-$version of Plancherel Theorem says that $$ \Vert f\Vert^2_2 =\frac{1}{\vert G \vert} \sum_{\pi \in \hat G^{\Phi}} d_{\pi} \Vert {\cal F} (f)(\pi) \Vert^2. $$ Therefore, $$ \Vert f\Vert_{\infty}^2 \leq \frac{1}{\vert G \vert} \Vert f \Vert^2 \, \sum_{\pi \in \hat G^{\Phi}} d_{\pi} . $$ Since $f$ is nonzero, we apply (*) to the above inequality and get the desired result. \endproof \begin{thebibliography}{20} \bibitem{eja} Galina, E. and Vargas, J., Eigenvalues and eigenspaces for the twisted Dirac operator over $SU(n,1)$ and $Spin(2n,1)$, Trans. Amer. Math. Soc., {\bf 345} (1994), 97-113. \bibitem{anna} Helversen-Pasotto, A., Repr\'esentation de Gelfand-Graev et identit\'es de Barnes, Enseign. Math. {\bf 32} (1986), 57-77. \bibitem{katz} Katz, N., Estimates for Soto-Andrade sums, J. reine angew. Math. {\bf 438} (1993), 143-161. \bibitem{arca} Soto-Andrade, J., Geometrical Gel'fand Models, tensor quotients and Weil representations, Proc. Symp. Pure Maths., {\bf 47}, AMS, Providence, 1987, 305-316. \bibitem{tesis}Soto-Andrade, J., R\'epresentations de certains groupes symplectiques, M\'em. {\bf 55-56}, Soc. Math. France, 1975. \bibitem{4} Varadarajan, Spherical functions, Springer, Berlin. \end{thebibliography} \end{document}