数理解析研究所講究録
第 792 巻 1992 年 144-150

144

A personal note on real Whittaker modelNov. 8, 1991

Takayuki ODA (RIMS, Kyoto)

In this note, I explain an “intuitive” understanding of the Jacquet integral for class-l

representation, in terms of the Helgason correspondence.
The contents of this note is substatially a hearsay from Hisayoshi Matsumoto andMasaki Kashiwara, and Toshio Oshima. I thank them. In order to make our “culture” up
to date, I wrote this.
The basic reference is $[K^{2}MO^{2}T]$ .

\S 1. Helgason correspondence
Let us recall basic ingredient on the Helgason correspondence. Let$G$

be a connected real

semisimple Lie group with finite center, $K$ a maximal compact subgroup of$G/K$. Thenis a Riemannian symmetric space. Let $P$ be a minimal parabolic subgroup, andlet $P=MAN$ be a Langlands decomposition of $P$ . We denote by
$A$

$G$

$\alpha$the Lie algebra of. Similary Germann letters represents the Lie algebras of the Lie group denoted by thecorresponding Latin letter. We denote by$\theta$

the Cartan involution with respect to theCartan decomposition.
1.1. Space of hyperfunctions on the boundaryLet$a_{C}^{*}$

be the complexification of the dual

denote by$L_{\lambda}$$e^{(\rho-\lambda)(\log a)}$of . For any linear form$a$$\log$We denote by$\mathcal{B}(G/P, L_{\lambda})$the space of all$L_{\lambda}$on$a_{C}^{*}$, we-valued hyperfunctions on $G/P$ . Thenis canonically identified with the space of all hyperfunctions

we put $\pi_{\lambda}(g)\psi(x)=\psi(g^{-1}x)$ , then$G$on: $Aarrow a$ is the inverse of the exponential map $exp:aarrow A$ .

$(x. man)=e^{(\lambda-\rho)(\log a)}\psi(x)(m\in M, a\in A, u\in N)$of$\lambda$

the line bundle on $G/P$ associated to the character of $P=MAN$ : $man\mapsto$. Here$\mathcal{B}(G/P, L_{\lambda})$$a^{*}$$\mathcal{B}(G/P, L_{\lambda})$.

$\psi$. For any $g\in G$ and$\pi_{\lambda}(g)\psi\in \mathcal{B}(G/P, L_{\lambda})$, and$\pi_{\lambda}$

on$G$

such that$\psi$$\psi\in \mathcal{B}(G/P, L_{\lambda})$,defines a representation

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1.2. Space of analytic eigenfunctions on $G/K$

Let $D(G/K)$ be the ring of invariant differential operators, and $A(G/K)$ be the space of
all analytic fumctions on $G/K$ .
For any algebra homomorphism $X$ of $D(G/K)$ into

$C$, we denote by $A(G/K;M(X))$

the space of ali analytic functions on $G/K$ satisfying the system $M(X)$ of the differentialequations:$\mathfrak{M}(X):\triangle u=X(\triangle)u$$(\triangle\in D(G/K))$.

For any $g\in G$ , and $f\in A(G/K)$ , we put $(\pi(g)f)(x)=f(g^{-1}x)$ . Then it is clearthat $A(G/K;M(X))$ is stable by $\pi(g)(g\in G)$ . Thus we obtain a representation of$G$

on$A(G/K;M(X))$ .

The space of K-invariant elements in $A(G/K;M(X))$ is one-dimensional (a fact ofspherical function), and there is a generator $\Phi_{X}\in A(G/K;\mathfrak{M}(X)^{K}$ normalized by thecondition $\Phi_{X}(eK)=1$ at the origin $eK\in G/K$ . Then for any $f\in \mathcal{A}(G/K;\mathfrak{M}(X))$$\int_{K}f(k^{-1}x)dk=f(eK)\Phi_{X}(x)$.

Here the Haar measure of $K$ is normalized such that $\int_{K}dk=1$ .
1. S. Matching of parameters

Let $W(A)$ be the Weyl group for $(g, a)$ . Then there is a natural identification$a_{C}^{*}/W(A)arrow\sim$$Hom(D(G/K), C)$ defined as follows:

Consider a direct sum decomposition:

.

$(*)$

Here$U(g_{C})=\overline{\mathfrak{n}}_{C}\cdot U(g_{C})+U(\alpha_{C})+U(g_{C})\cdot t_{C}$$\overline{\mathfrak{n}}_{C}=\theta(\mathfrak{n}_{C}),$$\theta$being the Cartan involution. Let$\#$: $U(a_{C})arrow U(g_{C})$ be the algebra

homomorphism induced by $\mathfrak{h}(H)=H-\rho(H)(\forall H\in a_{C})$ . Here$\rho$is half of the sum of

all positive roots. Let 6 be the projection $U(g_{C})arrow U(a_{C})$ with respect to the above

decomposition $(*)$ . Then the composition$\gamma$:$\# 0\delta$

induces an algebra isomorphism$D(G/K)arrow^{\sim}I_{W(A)}(\mathfrak{a}_{C})$,146
via the canonical inclusion$D(G/K)arrow U(g_{C})^{K}=\{Y\in U(g_{C})|Ad(k)\cdot Y=Y,\forall k\in K\}$.

Here $I_{W(A)}(ac)$ denotes the set of $W(A)$ -invariants in $U(a_{C})=S(a_{C})$ . Therefore we have$Hom(D(G/K), C)=Hom(Iw(A)(ac), C)$$=a_{C}^{*}/W(A)$For any A$\in a_{C}^{*}$, we denote byphism. The preimage ofWhen$\chi=\chi_{\lambda}$$\chi_{\lambda}$$\chi_{\lambda}$.

: $D(G/K)arrow C$ the corresponding algebra homomor-is given by $I_{\lambda}\{w\lambda|w\in W(A)\}$ .

, the function$\Phi_{\chi_{\lambda}}$is given by.

$\Phi_{\lambda}(gK)=\Phi_{\chi_{\lambda}}(gK)=\int_{K}e^{(\lambda-\rho)H(gk)}dk$

Here for $gEG,$ $H(g)E$$k\in K,$ $n\in N$$a$is the unique element such that $g=k\cdot(\exp H(g))\cdot n$ for some.

1.4. Poisson transformation
The poisson transformation$P_{\lambda}$

is defined as follows. For any$(P_{\lambda} \phi)(gK)=\int_{K}\phi(gK)dk$$\phi\in \mathcal{B}(G/P, L_{\lambda})$, we put.

Then we can show that$(P_{\lambda} \phi)(gK)=\int_{K}e^{-(\lambda+\rho)(H(g^{-1}k))}\phi(gK)dk$

Put$\Psi_{\lambda}(gK)=e^{-(\lambda+\rho)(H(g^{-1}))}$

can consider the restriction
show that$\psi_{K}$.

. Then $\psi_{\lambda}\in A(G/K;M(\chi))$ . For any$\psi_{K}$$\psi\in \mathcal{B}(G/P, L_{\lambda})$, we. Then $\psi_{K}(mk)=\psi\kappa(k)$ for any $m\in M,$ $k\in K$ . Thisis canonically identified with a hyperfunction on $K/M$ . Conversely for any

hyperfunction $\phi\in B(K/M)=\mathcal{B}(K)^{M}$ , we put$\psi(kan)=e^{(\lambda-\rho)(\log a)}\phi(k)$Clearly$\psi\in \mathcal{B}(G/P, L_{X})$$\mathcal{B}(K/M)$( $k\in K,$ $a\in A$ , and $n\in N$ ).

and $\psi_{K}=\phi$ . Thus we have an identification.

Via this identification,$P_{\lambda}$

is written as$(P_{\lambda} \phi)(gK)=\int_{K}\pi(k)\Psi_{\lambda}(gK)\phi(k)dk$

Hence$\mathcal{B}(G/P, L_{\lambda})\cong$$\triangle P_{\lambda}\phi=\chi_{\lambda}(\delta)\cdot P_{\lambda}\phi$which is elliptic,$P_{\lambda}\varphi$(for $\phi\in \mathcal{B}(K/M)$ ).

for all $\triangle\in D(G/K)$ . Since $D(G/K)$ contains the Laplacian,is real analytic.

147
Proposition. The Poisson transform$P_{\lambda}$

is a G-homomorphism of$\mathcal{B}(G/P),L_{\lambda})arrow \mathcal{A}(G/K;M(\chi))$

1.5. Boundary value map.

$\beta_{\lambda}$

We do not give the precise definition of$\beta_{\lambda}$. But we note the following: Let$M(\chi)$be thesystem of the differential equations defined by$(\triangle\in D(G/K))$$M(\chi):\triangle n=\chi(\triangle)n$Thenhas regular singularity along the edge $K/M\cross\{0\}$ . Moreover the roots of the$\mathfrak{M}(\chi)$indicial equation of$\mathfrak{M}(\chi)$coincide with

Anyway what is important for us is$\mathcal{B}(G/P;L_{\lambda})$$\{S(\lambda);\lambda\in I_{\chi}\}$$\beta_{\lambda}$.

gives a G-homorphism.

Under the as sumption$(A):- \frac{2\langle\lambda,\alpha\rangle}{\{\alpha,\alpha\rangle}\not\in N$$P_{\lambda}$

and$\beta_{\lambda}$.

,forare mutually inverse up to scalar multiple.

$\alpha\in\Sigma^{+}$,$\beta_{\lambda}$:$\mathcal{A}(G/K;\mathfrak{M}(\chi_{\lambda}))arrow$

148\S 2. Class-l Whittaker function
Lettobe the class-l principal series representation on a Hilbert space$(\pi_{\lambda}, H_{\lambda})$$\lambda\in \mathfrak{a}_{C}^{*}$. Choose a character$\psi$: $Narrow C^{*},$ an.d let

$\mu$

:

$H_{\lambda}$

associated$(\pi_{\lambda}, H_{\lambda}^{\infty})arrow C_{\psi}^{\infty}(N\backslash G)$

be theWhittaker functional.
Let$\Phi\in H_{\lambda}^{K}$

be the generator of$H_{\lambda}^{K}$. Put $w(g)=\mu(\pi_{\lambda}(g)\Phi)$ . Then$\forall g\in G,\forall n\in N$$\forall g\in G,\forall k\in K$,,$W(ng)=\psi(n)W(g)$ ;$W(nk)=W(g)$ .

Thus we can regard $W$ as a function on $G/K$ . Then $W$ belongs tous consider a subspace$A_{\psi}(N|G/K;\mathfrak{M}(X_{\lambda}))$of$\mathcal{A}(G/K;M(X_{\lambda}))$$A(G/K;\mathfrak{M}(X_{\lambda}))$, defined by$\{f(x)\in A(G/K;\mathfrak{M}(X_{\lambda}))|f(n\cdot x)=\psi(n)f(x)\}$Then$\beta_{\lambda}$

and$P_{\lambda}$.

induces an isomorphism$\mathcal{A}_{\psi}(N\backslash G/K;\mathfrak{M}(X_{\lambda}))\cong \mathcal{B}_{\psi}(N\backslash G/K;L_{\lambda})$Here. Let$\mathcal{B}_{\psi}(N\backslash G/P : L_{\lambda})$is a subspace of function$\phi\in \mathcal{B}(G/P : L_{\lambda})$$\phi(ng)=\psi(n)\phi(g)$Now let us consider the space$\mathcal{B}_{\psi}(N\backslash G/P:L_{\lambda})$$G=$$\cup$.
satisfying.

. Using Bruhat decomposition$NwP$.

$w\in W(A)$Hence$\phi\in \mathcal{B}_{\psi}(N\backslash G/P:L_{\lambda})$is determined by its “values” at $w\in W(A)$ . Hence$\dim_{C}\mathcal{B}_{\psi}(N\backslash G/P : L_{\lambda})=\dim_{C}A_{\psi}(N\backslash G/K;\mathfrak{M}(X_{\lambda}))=|W(A)|$Let$w_{0}$.

be the longest element of the Weyl group $W(A)$ . Then the Poisson Ivan formationof the constant function 1 on

$w_{0}$:$P_{\lambda}(1)$given an element ofshould be the Jacquet’s integral representation of .
$w$

2.1. Example$\mathcal{A}_{\psi}(G/K;\mathfrak{M}(X_{\lambda}))$. This149
Set $G=SL_{2}(R)$ . $P=\{(0* :) \in G\},$ $K=SO(2)$ . Then,$G=KP;G/P=KP/P$.
$B(G/P;L_{\lambda})\ni f$: $f(gP)=e^{\lambda-\rho_{p}}(p)f(g)$ .

Here $\rho_{P}=half$ sum of positive roots. Put $\psi(n)=\exp(2\pi icx)$ for $n=(_{0}^{1} x1)\in N$ , with afixed $c\in R$ . Then assume that$f$satisfies the conical condition:$f(ng)=\psi(n)f(g)$.

Write.

$p=(\begin{array}{ll}\sqrt{y} \sqrt{y}^{-1}X0 \sqrt{y}^{-1}\end{array})\in P$

Then$f(np)=\psi(n)e^{\lambda-\rho P}(p)=\exp(2\pi icx)(\sqrt{y})^{\lambda-1}$.

Identify $G/P\cong P^{1}(R)$ with’K, via$(\begin{array}{ll}cos\theta sin\theta-sin\theta cos\theta\end{array})(\begin{array}{l}10\end{array})=(\begin{array}{l}cos\theta-sin\theta\end{array})$&$(\cos\theta :-\sin\theta)\sim(-\cot\theta :1)$

inThen the Poisson transformation is written as follows for,$P^{1}(R)$.

$u\in \mathcal{B}(G/P, L_{\lambda})$.

$P_{\lambda}(u)(a)= \int_{K}u(ak)dk=\frac{1}{2\pi}\int_{-\pi}^{\pi}u(ar_{\theta})d\theta$Here write$ar_{\theta}=n_{1}wa_{2}n_{2},$ $i.e$$ar_{\theta}=(^{\sqrt{y}}$.

$\sqrt{y}^{0_{-1}}$$0$)$(\begin{array}{ll}cos\theta sin\theta-sin\theta cos\theta\end{array})=(\begin{array}{ll}l x_{1}0 1\end{array})(\begin{array}{ll}0 1-1 0\end{array})(\begin{array}{lll}s t0 s -1\end{array})$.

Then $s_{1}=-y\cot\theta$ . Set $u=f$ , then$P_{\lambda}(f)($($\sqrt{y}^{0_{-1}}$$0$)$)= \frac{1}{2\pi}\int_{-\pi}^{\pi}\exp(-2\pi icy\cot\theta)\cross(\sqrt{y}^{-1}\sin\theta)^{(\lambda-\rho_{P})}f(w)d\theta$$=f(w) \cdot y^{-(\frac{\lambda-}{2}\angle)}’\cdot\frac{1}{2\pi}\int_{-\pi}^{\pi}\exp(-2\pi icy\cdot\cot\theta)\cdot(\sin\theta)^{(\lambda-\rho_{P}+2)}\sin^{-2}\theta\cdot d\theta$.

150
Write $t=-\cot\theta$ . Then

$dt=\sim_{sjn\theta}d\theta\sin\theta=\sqrt{1+\cot\theta}^{1}$Hence$P_{\lambda}(f)( (\sqrt{y} \sqrt{y}))=f(w)\cdot y^{-(\frac{\lambda-\rho p}{2})}\int_{-\infty}^{\infty}\exp(2\pi icy\cdot t)(1+t^{2})^{-,(\lambda-\rho_{P})+2}dt1$

This is the Jacquet integral. General case is discussed similary.

RemarkThis correspondence of “algebraic Whittaker function” and Bruhat geometry$I$

seems to be valid in a very wide context. It seems applicable also four generalized Whittakermodel, or Gelfand-Graev representation. Wallach [W] already obtains some results in thisdirection.
RemarkUsing the contour integral representation of an element in$\Pi$$B_{\psi}(N\backslash G/P : L_{\lambda})$which has a support of lower dimension i.e. supp $(\psi)\subset N\backslash G/P-\{w_{0}P\}$ , then it seems
possible to have an integral representation of the all elements inwhen$\lambda$$A_{\psi}(N\backslash G/K;\mathfrak{M}(X_{\lambda}))$satisfies the assumption $(A)$ .

Reference

Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima and M. Tanaka,$[K^{2}MO\mathfrak{W}]$Eigenfunctions of invariant differential operators on a symmetic space, Ann. ofMath., 107 (1978), 1-39.

N. Wallach, Lie Algebra Cohomology and Holomorphic continuation of Generalized[W]$\backslash \ell_{-}^{(}\underline{\prime\urcorner\{)e}$$P_{c_{\mathscr{O}}}$

Jacquet Integrals, Advanced Studies in Pure Math., 14 (1988), 123-151.
). a1( Ht$\star\triangleright$$t_{y}$$cl_{\backslash }$$?\uparrow\geq$$j\Gamma$$|*4\uparrow^{}*ibi$$c_{\oint\infty el\infty\nu_{1}}-\ovalbox{\tt\small REJECT}\vee da\iota\wedge r_{G-\ovalbox{\tt\small REJECT}_{\mathfrak{U}^{A}}\nabla}$($H_{Y}^{\omega}$

w-&\mbox{\boldmath $\theta$}\iota-Y$pk_{\neg’}\nearrow$$kl_{\backslash _{(?,/<J}}(t/_{t^{K}},$$C_{Y}^{4\supset}(/\vee\backslash 6$$rightarrow k_{J/}tC\backslash \sim ce_{v\nearrow)}\gamma 3\not\supset\cdot\cdot\backslash ff|L\backslash ’arrow\succeq tB|f\sim \mathfrak{c}^{g_{P\S}}(i\{\vdash\lrcorner\Re\sigma$$C\uparrow^{)}e\sqrt[\mu le\lambda]{}\ovalbox{\tt\small REJECT}_{\vee 0}$$]f^{\Gamma}\alpha^{\sim}s)$.

$r_{tV}a$$\mathcal{O})^{\sim}B-\sigma_{R}^{<}\Gamma_{A}\sim\lambda,$[R.

$G-1\sqrt{}\neg$y.
I$1\sqrt 2$,p}$\backslash |_{\sim}^{\wedge}f- 7\nearrow_{c^{\neg}}\dagger s4_{\lambda}0$$G_{\theta}\circ l_{\wedge\zeta\infty\vee}c_{\sim}$

A い d.

\langle ($/ \int.$3?

)($\iota\ovalbox{\tt\small REJECT}\dot{\tau}\dot{a}k$

R. WW cA)

$3\star Y\backslash \cdot|A_{\circ}\ltimes\ h_{\tau’}$jC$/h_{C_{\urcorner}}(H_{\lambda\prime}^{\infty}c_{\gamma^{(}//\backslash \mathfrak{c}_{7})J}^{o}$L)$\mathfrak{u}\mathfrak{s}\theta^{\backslash }\star\}_{\alpha}u_{v}$.

$\backslash \cross Lkt\backslash 4$$cw_{\vee}^{\iota_{C}}\mathcal{A}4^{A_{C}}Larrow A_{\supset}$$/\ell farrow 27\theta$$(|\theta ff_{0})$ノ$t\cdot a^{\uparrow_{t\lambda}^{t}}\Omega\Lambda l\iota b_{\ell\triangleleft}^{l}$)$(ud \}_{qC^{J}}\int A_{S}b^{-}h_{c}^{f}r^{\mathfrak{p}}4\zeta t^{k\iota_{\vee J^{(}}c)\backslash p}, *r^{M\mathcal{E}rightarrow\ell_{1}}\cdot\not\simeq d’$$/\iota\alpha p_{YQ}\overline{u}F_{e;\infty^{r}}(t$$\gamma$

1Q} $?arrow/1I_{c}f$$L\sim^{\overline{\iota^{\backslash }}_{b11}}t1\overline{*}(7\llcorner(3$$F_{tLuc}f$

N.R.

]

$\square X$$2_{(P7}\sim 3$

l9$A9d\ovalbox{\tt\small REJECT}_{c}u^{J}m\nearrow\sim"$$c_{\gamma W}4\sim$$2_{C}$

も,友{’\alpha . w$/\prec \mathcal{U}$a7$/V_{d}102*$

4-欧/タヱ 33