\chapter{Laplacian Eigenvalues}

In this chapter we shall study the behavior of Laplacian eigenvalues under functoriality.
This is given by Theorem \ref{main laplace}, which we will later use in the proofs in Chapters 5 and 6.

Laplacian eigenvalues are obtained by evaluating the infinitesimal character on the
Casimir element. Hence we start by discussing the Casimir elements in Section \ref{lapl-casimir} and then consider the infinitesimal characters in Section \ref{laplace infin}. Finally, we prove Theorem \ref{main laplace} in Section \ref{laplace proof}.

Our main reference is \cite{knapp}, Chapter V.5. 

Throughout this chapter, all groups and representations are real, unless stated otherwise.

\section{Casimir Element}\label{lapl-casimir}

In this Section, we review the definition of the Killing form and Casimir element, following \cite{knapp} (mostly Chapter V).

Let $\lieg$ be a Lie algebra (viewed as a Lie subalgebra of $\mathrm{Mat}_{m\times m}(\mathbb C)$ for some $m$).
For $X, Y\in\lieg$, we define the Killing form as $B(X, Y)=\Tr(\mathrm{ad\ } X\ \mathrm{ad\ } Y)$. It is a symmetric bilinear form on $\lieg$. We also have the bilinear form $C(X, Y)=\Tr(XY)$.
When $\lieg$ is a simple complex Lie algebra, $B(X, Y)=b_\lieg C(X, Y)$ for some constant $b_\lieg$. 

We shall be dealing mostly with the Lie algebras $\liesl_N(\mathbb C)$, $\liesp_{2n}(\mathbb C)$, $\lieso_N(\mathbb C)$, and $\lieso_{2n}^\ast(\mathbb C)$.
For each of them, the Killing form is a multiple of $C(X, Y)$ and the corresponding constants are 
$b_{\liesl(N)}=2N$, $b_{\liesp(2n)}=2n+2$, and $b_{\lieso(N)}=b_{\lieso^\ast(N)}=N-2$ (see eg. \cite{fh}, Ex. 14.36).

Now, let $\lieg$ be a real Lie algebra and $\liegc=\lieg\otimes_\mathbb R\mathbb C$ its complexification.
Let $\lieh$ be a Cartan subalgebra of $\lieg$ (and $\liehc$ its complexification).
Let $\Phi$ be the set of roots of $\liegc$ with respect to $\liehc$.

Choose an orthonormal basis $H_1, \dots, H_n$ of $\liehc$ with respect to the Killing form $B$ on $\liegc$. Also choose root vectors $E_\alpha$ so that $B(E_\alpha, E_{-\alpha})=1$ for all $\alpha\in\Phi$. Then the Casimir element of $\liegc$ is
$\Omega=\sum_{i=1}^n H_i^2 + \sum_{\alpha\in\Phi} E_\alpha E_{-\alpha}$ as an element of
the center $Z(\liegc)$ of the universal enveloping algebra.

We have the Harish-Chandra isomorphism $\gamma: Z(\liegc)\rightarrow \mathcal H^W$, where
$\mathcal H$ is the universal enveloping algebra of $\liehc$ and $\mathcal H^W$ are its 
Weyl group invariants. For more details, see \cite{knapp}, Ch. V.5.

To compute the Laplacian eigenvalues of our representations, we shall need the images of the Casimir element under the Harish-Chandra isomorphism. In general,
it is $\gamma(\Omega)=\sum H_i^2 - |\delta|^2$, where $\delta$ is half of the sum of positive roots. 


\section{Infinitesimal Character}\label{laplace infin}

Let $G$ be a Lie group and let $\lieg$, $\lieh$, $\liegc$, and $\liehc$ be the corresponding Lie algebras as above.

For $\mu\in\liehc^*$ define $\chi_\mu(Z)=\mu(\gamma(Z))$ for $Z\in Z(\liegc)$. 
Note that $\gamma(Z)\in U(\liehc)=\mathcal H$, 
and so it makes sense to evaluate $\mu$ on $\gamma(Z)$ -- here we are of course extending $\mu$ to a character of the universal enveloping algebra $\mathcal H$. It follows from the Harish-Chandra isomorphism that all characters of $Z(\liegc)$
are of this form.

Let $\pi$ be an irreducible admissible representation of $G$.
From $\pi$ we can obtain a representation of $Z(\liegc)$. This representation is then a character, called the
infinitesimal character of $\pi$ and denoted by $\chi_\pi$. 
This character is of the form
$\chi_\pi=\chi_\mu$ for some $\mu\in\liehc^*$ as above. This $\mu$ is uniquely determined up to the action of $W$ and it is sometimes also called the infinitesimal character of $\pi$.

The Laplacian eigenvalue of $\pi$ is then obtained by evaluating the infinitesimal character on the Casimir element, i.e., $\chi_\pi(\Omega)$. 


\section{Behavior under Functoriality}\label{laplace proof}

To see what happens to the Laplacian eigenvalue under functoriality, we need to
study what happens to the infinitesimal character.

Let $W_\mathbb R$ be the Weil group. By the local Langlands correspondence, to each $\pi$ corresponds a parameter 
$\varphi: W_\mathbb R\rightarrow$ $^L G$, where $^L G$ is the Langlands dual group. By conjugating $\varphi$ if necessary, we may assume that 
$\varphi(W_\mathbb R)\subset N(^L H^0)$, where $H$ is a maximal torus in $G$. Then $\varphi$ is of the form 
$\varphi(z)=z^\mu\overline z^{\mu^\prime}$ for $z\in\mathbb C^\times\subset W_\mathbb R$, 
where $\mu, \mu^\prime\in X_*(H^\vee)\otimes \mathbb C=X^*(H)\otimes \mathbb C$ are such that $\mu-\mu^\prime\in X^*(H)$. Here $\varphi(z)=z^\mu\overline z^{\mu^\prime}$ stands for 
$\varphi(e^s)=\exp(s\mu+\overline s\mu^\prime)\in$ $^L H^0$ for $s\in\mathbb C$.

By definition we have $X_*(H^\vee)=X^*(H)$, and so we can view $\mu$ as an element of $X^*(H)\otimes \mathbb C$ and attach to it a character $\chi_\mu$ of
$Z(\liegc)$ via the Harish-Chandra homomorphism. If $\varphi$ is the Langlands parameter of $\pi$, then $\chi_\mu$ is the infinitesimal character of $\pi$.
(See for example \cite{av} for details.) 

\

In the rest of this section, let $G=\mathbf G(\mathbb R)$ be a quasisplit classical group. 
Let $\Pi$ be a self-dual irreducible admissible representation of $GL_N(\mathbb R)$ which is a functorial lift of an 
irreducible admissible representation $\pi$ of $G$. The Laplacian eigenvalue of $\Pi$ is determined from the infinitesimal character of the restriction of $\Pi$ to $SL_N(\mathbb R)$, and so we will be dealing with $SL$ and $\liesl$.

Let $\varphi: W_\mathbb R\rightarrow$ $^L G$ be the Langlands parameter of $\pi$. The Langlands parameter of $\Pi$ is then given by composition with the
inclusion $^L G\hookrightarrow$ $^L SL_N$.

\begin{proposition}\label{functoriality infin}
Let $H$ be a maximal torus of $G$ and $T$ a maximal torus of $SL_N(\mathbb R)$ such that $H^\vee\hookrightarrow T^\vee$. 
This induces an inclusion $\iota: X^*(H)\otimes \mathbb C=X_*(H^\vee)\otimes \mathbb C\hookrightarrow X_*(T^\vee)\otimes \mathbb C=X^*(T)\otimes \mathbb C$.

Then the infinitesimal character of $\Pi$ is $\iota(\mu)$, where $\mu$ is the infinitesimal character of $\pi$.
\end{proposition}

\begin{proof}
Immediately follows from the discussion at the beginning of this section.
\end{proof}

\begin{theorem}\label{main laplace}
Let $\lambda$ and $\Lambda$ be the Laplacian eigenvalues of $\pi$ and $\Pi$, respectively.
Then there are constants $c_G>0$ and $d_G$ such that $\Lambda=c_G\lambda+d_G$.

We have $c_{Sp(2n)}=\frac{2n+2}{2n+1}$, $c_{SO(2n)}=c_{SO^\ast(2n)}=\frac{n-1}{n}$, 
$c_{SO(2n+1)}=\frac{2n-1}{2n}$.
\end{theorem}

\begin{proof}
To prove the theorem, we evaluate the infinitesimal characters on Casimir elements using
Proposition \ref{functoriality infin}.

Denote by $E_{ij}$ the matrix with 1 at the position $(i, j)$ and 0 elsewhere and denote $E_{ii}$ also by
$E_i$. Let $E_i^\ast\in\lieh^\ast$ be the corresponding dual element. 
For $A=\sum a_iE_i\in\liehc$, we define $A^\ast=\sum a_iE_i^\ast\in\liehc^\ast$.

\

For $\lieg=\liesl_N(\mathbb R)$, let $\liea$ be the subalgebra of diagonal matrices
with trace zero. Extend the set $E_i-E_{n+i}$ to a basis of $\lieac$ (where $N=2n+\varepsilon$ with $\varepsilon= 0, 1$).
We have $B(E_i-E_{n+i}, E_j-E_{n+j})=2N\Tr((E_i-E_{n+i})(E_j-E_{n+j}))=2N\delta_{ij}$ (where $\delta_{ij}$ is Kronecker delta). 
Hence we can extend $A_i=\frac 1{\sqrt{2N}}(E_i-E_{n+i})$ to an orthonormal basis of
$\lieac$ with respect to the Killing form $B$.

\

Take $G=Sp_{2n}(\mathbb R)$. Take $\liehc$ to be the subalgebra of diagonal matrices in $\lieg$;
we can choose $b_i=E_i-E_{n+i}$ as a basis of $\liehc$.
We have $B(b_i, b_j)=(2n+2)\Tr(b_ib_j)=2(2n+2)\delta_{ij}$. 
Thus $H_i=\frac 1{\sqrt{2(2n+2)}}b_i$ form an orthonormal basis of $\liehc$ with respect to the Killing form $B$.

Let $\pi$ be a representation of $G$ with infinitesimal character $\mu=(\mu_1, \dots, \mu_n)\in\liehc^\ast$ with respect to the basis $b_i^\ast=E_i^\ast-E_{n+i}^\ast$, i.e., 
$\mu=\sum \mu_i b_i^\ast$. We have $\gamma(\Omega)=\sum H_i^2 - |\delta_{Sp}|^2$
and $b_i^\ast(H_j)=\frac 2{\sqrt{2(2n+2)}}\delta_{ij}$, and so
we see that $$\mu(\gamma(\Omega))=\sum\mu_i^2b_i^\ast(H_i)^2-|\delta_{Sp}|^2=
\frac 2{2n+2}\sum\mu_i^2-|\delta_{Sp}|^2.$$


Let $\Pi$ be the restriction of the functorial lift of $\pi$ to $SL_{2n+1}(\mathbb R)$.
The infinitesimal character of $\Pi$ is then $\iota(\mu)=(\mu_1, \dots, \mu_n,-\mu_1, \dots,-\mu_n)\in\lieac^\ast$. 
Note that $\iota(\mu)=\sum\mu_i (E_i-E_{n+i})^\ast$.
Thus the Laplacian eigenvalue
$$\iota(\mu)(\gamma(\Omega))=\sum\mu_i^2(E_i-E_{n+i})^\ast(A_i)^2-|\delta_{SL}|^2=
\frac 2{2n+1}\sum\mu_i^2-|\delta_{SL}|^2.$$ We see that $c_{Sp(2n)}=\frac{2n+2}{2n+1}.$

\

Take $G=SO_{2n}(\mathbb R)$. This is entirely analogous to the case of $Sp_{2n}$,
the only differences are that $B(b_i, b_j)=(2n-2)\Tr(b_ib_j)$ and that $\pi$ lifts to $SL_{2n}(\mathbb R)$, which leads to 
$c_{SO(2n)}=\frac{2n-2}{2n}=\frac{n-1}{n}$.

\ 

The case of $G=SO_{2n+1}(\mathbb R)$ is also analogous, we get 
$c_{SO(2n+1)}=\frac{2n-1}{2n}$.

\

Let us finally discuss the case when $G=SO^\ast_{2n}(\mathbb R)$ in more detail.
Take $G$ and $\lieg$ to be defined with respect to the matrix $M=(m_{ij})$ with 
$m_{ij}=1$ for $i+j=2n$, $1\leq i\leq n-1$, or $n\leq i=j\leq n+1$, and $m_{ij}=0$ otherwise.
Choose the Cartan algebra $\lieh$ to be the span of $b_j=E_j-E_{2n-j}$ for $1\leq j\leq n-1$, 
and $b_n=E_{n, n+1}-E_{n+1, n}$. Note that in this case the Killing form is not 
positive definite on $\lieh$, as we have $B(b_n, b_n)=-2$.
Setting $H_j=\frac 1{\sqrt{2(2n-2)}}b_j$ we get that $$\gamma(\Omega)=\sum_{j=1}^{n-1} H_j^2 - H_n^2 -|\delta_{SO^\ast}|^2.$$

Let $\pi$ be a representation of $G$ with infinitesimal character $\mu=(\mu_1, \dots, \mu_n)\in\liehc^\ast$ with respect to the basis $b_j^\ast$. 
Then $$\mu(\gamma(\Omega))=\sum_{j\leq n-1}\mu_j^2b_j^\ast(H_j)^2-\mu_n^2b_n^\ast(H_n)^2 -|\delta_{SO^\ast}|^2=
\frac 2{2n-2}(\sum_{j\leq n-1}\mu_j^2-\mu_n^2)-|\delta_{SO^\ast}|^2.$$

Let $\Pi$ be the restriction of the functorial lift of $\pi$ to $SL_{2n}(\mathbb R)$ and let $\varphi: W_\mathbb R\rightarrow$ $^L G$ be the Langlands parameter of $\pi$.
Viewing $\mu$ as a cocharacter of $H^\vee$ we get that $\mu(\mathbb C^\times)$ is not in the torus $^LT^0$ -- we have to conjugate the part corresponding to $H_n$, originally of the form $\left(\begin{array}{cc} \cos\mu_n & \sin\mu_n \\ -\sin\mu_n & \cos\mu_n \end{array}\right)$, which we conjugate to $\left(\begin{array}{cc} e^{i\mu_n} & 0 \\ 0 & e^{-i\mu_n} \end{array}\right)$ (here $i=\sqrt{-1}$).
This gives us that the infinitesimal character of $\Pi$ is then $\iota(\mu)=(\mu_1, \dots, i\mu_n,-i\mu_n, \dots,-\mu_1)\in\lieac^\ast$. 
Hence
$$\iota(\mu)(\gamma(\Omega))=
\frac 2{2n}(\sum\mu_j^2-\mu_n^2)-|\delta_{SL}|^2.$$ We conclude that
$c_{SO^\ast(2n)}=\frac{2n-2}{2n}=\frac{n-1}{n}$.

\

This concludes the proof of Theorem \ref{main laplace}. Note that one can also easily compute the constants $d_G$ by calculating the various values $|\delta|^2$.
\end{proof}
{"config": "arxiv", "file": "1406.0385/ch3.tex"}