A Construction of Representations of Loop Group and Affine Lie Algebra of $\mathfrak{sl}_n$

TL;DR

This paper constructs a new representation of the loop group and affine Lie algebra of sl_n, providing explicit formulas and a realization of Whittaker functionals, advancing the understanding of their structure and representations.

Contribution

It introduces a novel construction of loop group and affine Lie algebra representations with explicit formulas and realizations of Whittaker functionals.

Findings

Explicit representation formulas for loop group and affine Lie algebra of sl_n.

Concrete realization of Whittaker functionals in the dual representation.

Advancement in understanding the structure of affine Lie algebra representations.

Abstract

In this paper, we construct a representation of loop group and derive the formula of the corresponding representation of the affine Kac-Moody algebra with level 1. And we also provide a concrete realization of Whittaker functionals in the dual representation.

Equations

$$G\stackrel{\mathrm{def}}{=}\{a(t)\in \mathbb{R}((t){)}^{\ast }|\mathrm{the}\mathrm{leading}\mathrm{coefficient}\mathrm{of}a(t)\mathrm{is}\pm 1\}.$$

$$\pi (n)f(x)=\sum _{i=1}^{n}f(\frac{x}{n}+\frac{i}{n}).$$

$$\pi (g)f(x)=\sum _{r\in R^n/g{R}^n}f(g^{-1}(x+r))=\sum _{r\in g^{-1}{R}^n/R^n}f(g^{-1}x+r).$$

$$V_{\lambda }=\{f\in Fun(F^n/R^n)|\pi (\beta I_n)f=\lambda f\}.$$

$$\pi (g,\mu )f(x)\stackrel{\mathrm{def}}{=}{\int }_{g^{-1}{R}^n/R^n}f(g^{-1}x+v)\mu (dv).$$

$$F=\mathbb{R}((t)),R=\mathbb{R}[[t]],R_{-}=\mathbb{R}[t^{-1}]t^{-1},F=R\oplus R_{-}.$$

$${\dim }_{\mathbb{R}}{g}^{-1}{R}^n/R^n=N.$$

$$V_g=g^{-1}{R}^n/R^n.$$

$$\pi (g,\mu )f(x):={\int }_{V_g}f(g^{-1}x+y)\mu (dy).$$

$$\pi (g,\mu )f(x+a)={\int }_{V_g}f(g^{-1}x+g^{-1}a+y)\mu (dy)$$

$${\int }_{V_g}f(g^{-1}x+g^{-1}a+y)\mu (dy)={\int }_{V_g}f(g^{-1}x+y)\mu (dy)=\pi (g,\mu )f(x)$$

$$(g_1,{\mu }_1)(g_2,{\mu }_2)=(g_1{g}_2,{\mu }_1\ast {\mu }_2).$$

$$0\to g_2^{-1}{R}^n/R^n\to g_2^{-1}{g}_1^{-1}{R}^n/R^n\to g_2^{-1}{g}_1^{-1}{R}^n/g_2^{-1}{R}^n\to 0$$

$${\int }_{V_{g_1{g}_2}}f(x)({\mu }_1\ast {\mu }_2)(dx)$$

$${\int }_{V_{g_1{g}_2{g}_3}}f(v)({\mu }_1\ast ({\mu }_2\ast {\mu }_3))(dv)$$

$$=$$

$$\pi (g_1{g}_2,{\mu }_1\ast {\mu }_2)f(x)=$$

$$=$$

$$\pi (g_1,{\mu }_1)(\pi (g_2,{\mu }_2)f)(x)=$$

$$=$$

$$\pi (u,{\mu }_{st})f(x)=f(u^{-1}x).$$

$$t^{\mathbf{k}}=\mathrm{diag}(t^{k_1},\cdots ,t^{k_m})$$

$$V_{t^{\mathbf{k}}}=\mathrm{Span}(t^{-i_j}{e}_j|1\le j\le m,1\le i_j\le k_j),$$

$$(uv,{\mu }_{\mathrm{st}})=(u,{\mu }_{\mathrm{st}})(v,{\mu }_{\mathrm{st}}),$$

$$(t^{\mathbf{k}+{\mathbf{k}}^{\prime }},{\mu }_{\mathrm{st}})=(t^{\mathbf{k}},{\mu }_{\mathrm{st}})(t^{{\mathbf{k}}^{\prime }},{\mu }_{\mathrm{st}}).$$

$$(t^k{I}_n,{\mu }_{st})(c{I}_n,{\mu }_{st})=|c{|}^{kn}(c{I}_n,{\mu }_{st})(t^k{I}_n,{\mu }_{st}).$$

$$(t^k{I}_n,{\mu }_{st})(u,{\mu }_{st})=|\det (u{|}_{t=0}){|}^k(u,{\mu }_{st})(t^k{I}_n,{\mu }_{st}).$$

$$\pi (t{I}_n,{\mu }_{\mathrm{st}})\pi (c{I}_n,{\mu }_{st})f(x)$$

$${\pi }_{t^k}=\pi (t^k{I}_n,{\mu }_{\mathrm{st}}),{\pi }_c=\pi (c{I}_n,{\mu }_{\mathrm{st}}).$$

$$\mathcal{S}(R_{-}^n{)}_{\lambda }=\mathcal{S}({\mathbb{R}}^n[t^{-1}]t^{-1}{)}_{\lambda }=\{f\in S(R_{-}^n)|{\pi }_{t}f=\lambda f\}.$$

$$G{L}_n(F{)}_0=\{g\in G{L}_n(F)|\det g\text{ has the leading coefficient }\pm 1\}$$

Full text

A Construction of Representations of Loop Group and Affine Lie Algebra of $\mathfrak{sl}_{n}$

Xuanzhong Dai, Yongchang Zhu

Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

mazhu@ust.hk, xdaiac@connect.ust.hk

The research is supported by Hong Kong RGC grant 16305715.

1. Introduction

The loop groups of simple Lie groups and their algebraic counterparts affine Kac-Moody algebras are infinite dimensional generalizations of simple Lie groups and simple Lie algebras. Many results of latter objects have infinite dimensional generalizations. In this work we construct a family of representations of certain central extension of loop group $SL_{n}({\mathbb{R}}((t)))$ and its affine Lie algebra $\widehat{\mathfrak{sl}}_{n}$. This family is parametrized by the characters of the multiplicative group

[TABLE]

The group $G$ is a subgroup of a certain central extension of loop $GL_{1}({\mathbb{R}}((t)))$. Our construction can be interpreted loosely as an affine analog of the theta correspondence for the dual pair $GL_{1}$ and $GL_{n}$. In [Z], the second author constructed a Weil representation for metaplectic loop group of $SP_{2n}$, in which the above group $G$ and the lifting of $SL_{n}({\mathbb{R}}((t)))$ form a commuting pair. Our construction can be viewed as the theta lifting of characters of $G$ to representations of the central extension of $SL_{n}({\mathbb{R}}((t)))$. Since this connection is formal and the calculations are rather involved, we choose to present our construction in a way independent of the work in [Z].

The underline space of our representation consists of functions on ${\mathbb{R}}^{n}((t))$ that is periodic under the translation by elements in ${\mathbb{R}}^{n}[[t]]\subset{\mathbb{R}}^{n}((t))$, so it is a certain subspace of functions on ${\mathbb{R}}^{n}((t))/{\mathbb{R}}^{n}[[t]]={\mathbb{R}}^{n}[t^{-1}]t^{-1}$. The operators appear in our representation are integral operators. The operators for the corresponding representation of affine Lie algebras are infinite sums of quadratic differential operators, which is similar to early works of free field realizations [Fr] [KP] [FF], but the operator for a negative mode element in $\widehat{\mathfrak{sl}}_{n}$ has one term as an integral operator. And the representations we obtain are not highest weight modules or in the category ${\cal O}$.

To introduce our construction, we begin with two similar but much simpler cases. Let $C({\mathbb{R}}/{\mathbb{Z}})$ be the space of continuous functions on the circle ${\mathbb{R}}/{\mathbb{Z}}$. Although the multiplicative semigroup ${\mathbb{Z}}_{\neq 0}$ doesn’t act on ${\mathbb{R}}/{\mathbb{Z}}$ in any natural way, it acts on the function space $C({\mathbb{R}}/{\mathbb{Z}})$ as Hecke operators: for a non-zero integer $n$, let $\pi(n)$ be the operator given by

[TABLE]

It is easy to check that $\pi(m)\pi(n)=\pi(mn)$. The common eigenfunctions for $\pi(n)$’s with certain continuity properties are classified in [M].

The second case is related to representations of $GL_{n}(F)$, where $F$ is a non-Archmedean local field. Let $R$ be the ring of integers of $F$. We consider the space $Fun(F^{n}/R^{n})$ of complex valued functions on $F^{n}$ that is periodic with period $R^{n}$. Let $M_{n}^{\prime}(R)$ be the semi-group of $n\times n$-matrices with entries in $R$ and with non-zero determinant. For $g\in M_{n}^{\prime}(R)$ and $f(x)\in Fun(F^{n}/R^{n})$, we view $f$ as a function on $F^{n}$ which is invariant under the translations by $R^{n}$. The function $f(g^{-1}x)$ is periodic with periods $gR^{n}$. Since $g$ has entries in $R$, $gR^{n}\subset R^{n}$, and the quotient space $R^{n}/gR^{n}$ is a finite set. To get a periodic function with period $R^{n}$, we take the average of $f(g^{-1}x)$ over $R^{n}/gR^{n}$, we get linear operator $\pi(g):Fun(F^{n}/R^{n})\to Fun(F^{n}/R^{n})$ :

[TABLE]

It is easy to prove that $\pi(g_{1})\pi(g_{2})=\pi(g_{1}g_{2})$. So we have a representation of semi-group $M_{n}^{\prime}(R)$ on $Fun(F^{n}/R^{n})$. To get a representation of $GL_{n}(F)$, we invert the operator $\pi(\beta I_{n})$, where $\beta\in R$ is a prime. A direct way is to consider the space

[TABLE]

For $g\in GL_{n}(F)$, we decompose it as $g=\beta^{k}g^{\prime}$, where $k\in{\mathbb{Z}}$, $g^{\prime}\in M_{n}^{\prime}(R)$, then $\pi_{\lambda}(g)=\lambda^{k}\pi(g^{\prime})$ is independent of the decomposition, and we have $\pi_{\lambda}(g_{1}g_{2})=\pi_{\lambda}(g_{1})\pi_{\lambda}(g_{2})$. In general, for a rational morphism of a reductive groups $G\to GL_{n}$, then $V_{\lambda}$ is a representation of $G$ via the pull-back. In particular, we have an embedding $GL_{n}\times GL_{n}\to GL_{n^{2}}$ by the action $(g_{1},g_{2})x=g_{1}xg_{2}^{t}$, where $x$ is an $n\times n$ matrix. Our representation $V_{\lambda}$ of $GL_{n}\times GL_{n}$ can be used to give an interpretation of gamma factors of irreducible admissible representations of $GL_{n}(F)$ as defined in [GJ]. We will discuss this connection in a separate work.

The field ${\mathbb{R}}((t))$ and a non-Archedemean local field have many similarities. The above construction generalizes to $SL_{n}$ over ${\mathbb{R}}((t))$. From now on, we denote $F$ the field ${\mathbb{R}}((t))$, and $R$ the subring ${\mathbb{R}}[[t]]$. Let $M_{n}^{\prime}(R)=M_{n}^{\prime}({\mathbb{R}}[[t]])$ be the semigroup of $n\times n$ matrices over $R$ with non-zero determinant. We follow the construction (1.3). For a function $f$ on $Fun(F^{n}/R^{n})$, i.e., $f$ is a function on $F^{n}$ with periods $R^{n}$, and $g\in M_{n}^{\prime}(R)$. $f(g^{-1}x)$ is periodic with periods $gR^{n}$. But now $R^{n}/gR^{n}\cong g^{-1}R^{n}/R^{n}$ is not finite, it is a finite dimensional vector space over ${\mathbb{R}}$. To generalize (1.3), we choose a Haar measure $\mu$ on $g^{-1}R^{n}/R^{n}$, we introduce operator $\pi(g,\mu)$ by

[TABLE]

The collection of all pairs $(g,\mu)$ form a semigroup $\widehat{M_{n}}(R)^{\prime}$ that is a central extension of $M_{n}^{\prime}(R)$. We can again consider the eigenspace of $\pi(tI_{n},\mu)$. Now the situation is different from the local field case in that $\pi(tI_{n},\mu)$ don’t commutes with $\pi(g,\mu)$ for general $g\in M_{n}^{\prime}(R)$, it only commutes with operators $\pi(g,\mu)$ with ${\rm det}\,g$ has leading coefficient $\pm 1$, as a result, the eigenspace is a representation of a subgroup of $\widehat{GL_{n}}(F)$ that contains $\widehat{SL_{n}}(F)$. We will study various properties of this representation, and derive a formula for the corresponding representation of affine Kac-Moody algebra of $\mathfrak{sl}_{n}$.

The structure of this paper is as follows. In Section 2, we will give details of the construction of the representation of loop group $\widehat{SL_{n}}({\mathbb{R}}((t)))$ mentioned above and study its properties. In Section 3, we define a “maximal compact subgroup” for $\widehat{SL_{n}}(\mathbb{R})$ and construct a Gaussian function fixed by it. In Section 4, we derive the formula of the corresponding representation of the affine Kac-Moody algebra $\widehat{\mathfrak{sl}_{n}}$, and this representation has level $1$. In Section 5, we consider the dual action of $\widehat{\mathfrak{sl}_{n}}$ and construct various highest weight representations of level $-1$. In Section 6. we construct a representation of $\widehat{\mathfrak{sl}_{n}}\oplus\widehat{\mathfrak{sl}_{n}}$ via embedding $GL_{n}\times GL_{n}\to GL_{n^{2}}$, the representation has the special property that the level is critical. And we construct Whittaker functionals in the representation.

We wish to thank I.Frenkel for discussions.

2. A Construction of a Representation of Loop $SL_{n}$.

We continue to use the notations

[TABLE]

Recall that a function $f$ on a finite dimensional vector space over ${\mathbb{R}}$ is called a Schwartz function if all its partial derivatives $\partial^{I}f$ of arbitrary order is rapidly decay in the sense that for all $m$, $(1+|x|^{m})\partial^{I}f(x)$ is bounded. A key property of the space ${\cal S}(V)$ of all Schwartz functions on $V$ is that it is closed under taking derivatives, multiplying by polynomials and taking partial Fourier transforms.

For an infinite dimensional vector space $V$ over ${\mathbb{R}}$ such as $V=R_{-}^{n}={\mathbb{R}}^{n}[t^{-1}]t^{-1}$, a function $f$ on $V$ is called a Schwartz function if the restriction of $f$ to every finite dimensional space is a Schwartz function. We denote by ${\cal S}(F^{n}/R^{n})={\cal S}(R_{-}^{n})$ the space of all Schwartz functions on $R_{-}^{n}={\mathbb{R}}^{n}[t^{-1}]t^{-1}$.

Let $M_{n}^{\prime}(R)$ be the semi-group of $n\times n$ matrices over $R$ with non-zero determinant. We first construct a representation of a central extension of $M_{n}^{\prime}(R)$ on ${\cal S}(F^{n}/R^{n})$.

Let $\pi_{-}:F^{n}\to R_{-}^{n}$ be the projection map with respect to the decomposition $F^{n}=R^{n}\oplus R_{-}^{n}$. A function $f\in S(R_{-}^{n})$ gives a function on $F^{n}$, which we still denote by $f$, by the pull-back along $\pi_{-}$, so $f(x)=f(\pi_{-}(x))$. It is clear that $f$ is $R^{n}$-periodic. Conversely, $R^{n}$-periodic on $F^{n}$ is the pull-back for a unique function on $R_{-}^{n}$. We will not distinguish the $R^{n}$-periodic functions on $F^{n}$ and functions on $R_{-}^{n}$.

For each $g\in M_{n}^{\prime}(R)$ and an $R^{n}$-periodic function $f$ on $F^{n}$, $f(g^{-1}x)$ is $gR^{n}$-periodic. Since $g$ has entries in ${\mathbb{R}}[[t]]$, we have $gR^{n}\subset R^{n}$, therefore $R^{n}\subset g^{-1}R^{n}$. The quotient space $g^{-1}R^{n}/R^{n}$ is a finite dimensional space over ${\mathbb{R}}$. If ${\rm det}\,g=t^{N}(c_{0}+c_{1}t+\cdots)$, $c_{0}\neq 0$, then

[TABLE]

We denote

[TABLE]

Let $\mu$ be a Haar measure on $V_{g}$, then we consider the operator $\pi(g,\mu)$ given by

[TABLE]

For $a\in R^{n}$,

[TABLE]

since $g^{-1}a\in g^{-1}R^{n}$, we change variable $g^{-1}a+y\to y$, using the translation variance of the measure $\mu$, we see that

[TABLE]

So $\pi(g,\mu)f(x)$ is $R^{n}$-periodic. Since a linear change of variable of a Schwartz function is again a Schwartz function so $f(g^{-1}x)$ is a Schwartz function. And if $h(x,y)$ is a Schwartz function on $(x,y)\in U\times V$ for two finite dimensional spaces $U$ and $V$, then $\int_{V}h(x,y)dy$ is a Schwartz function on $x\in U$ for any Haar measure $dy$ on $V$, we see that $\pi(g,\mu)f(x)\in{\cal S}(R_{-}^{n})$.

Let $\widehat{M_{n}}(R)^{\prime}$ be the set of pairs $(g,\mu)$, where $g$ is an element in $M_{n}(R)^{\prime}$, and $\mu$ is a Haar measure on $V_{g}$. We define a multiplication on $\widehat{M_{n}}(R)^{\prime}$ so that $\pi(g,\mu)$ is a representation on ${\cal S}(R_{-}^{n})$. For $(g_{1},\mu_{1}),(g_{2},\mu_{2})\in\widehat{M_{n}}(R)^{\prime}$,

[TABLE]

where $\mu_{1}\ast\mu_{2}$ is the Haar measure on $V_{g_{1}g_{2}}$ given as follows. The chain $g_{1}g_{2}R^{n}\subset g_{1}R^{n}\subset R^{n}$ induces the chain $R^{n}\subset g_{2}^{-1}R^{n}\subset g_{2}^{-1}g_{1}^{-1}R^{n}$. Let $\mu_{1}^{\prime}$ be the pushforward measure on $\tilde{V}=g_{2}^{-1}g_{1}^{-1}R^{n}/g^{-1}_{2}R^{n}$ of $\mu_{1}$ under the isomorphism $\phi:V_{g_{1}}=g_{1}^{-1}R^{n}/R^{n}\to g_{2}^{-1}g_{1}^{-1}R^{n}/g^{-1}_{2}R^{n}$, sending $y\;({\rm mod}\,R^{n})$ to $g_{2}^{-1}y\;({\rm mod}\,g^{-1}_{2}R^{n})$. Consider the short exact sequence:

[TABLE]

where the second arrow refers to the inclusion map, and the third arrow refers to the corresponding quotient map. Then we define the Haar measure $\mu_{1}\ast\mu_{2}$ on $V_{g_{1}g_{2}}=g_{2}^{-1}g_{1}^{-1}R^{n}/R^{n}$ such that, for $f\in L^{1}(g_{2}^{-1}g_{1}^{-1}R^{n}/R^{n})$,

[TABLE]

Proposition 2.1**.**

$\widehat{M_{n}}(R)^{\prime}$* is a semi-group under the product (2.3). (2.2) gives a representation of $\widehat{M_{n}}(R)^{\prime}$ on ${\cal S}(R_{-}^{n})$.*

Proof. It is clear that $V_{I_{n}}=\{0\}$, and let $\mu_{st}$ denote the counting measure on $V_{I_{n}}$. Obviously $(I_{n},\mu_{st})\in\widehat{M_{n}}(R)^{\prime}$ is the identity element . To prove (2.3) is associative for $(g_{1},\mu_{1}),(g_{2},\mu_{2}),(g_{3},\mu_{3})$, let $f$ be arbitrary integrable function on $V_{g_{1}g_{2}g_{3}}=g_{3}^{-1}g_{2}^{-1}g_{1}^{-1}R^{n}/R^{n}$, from the definition (2.4) of $\ast$, we have

[TABLE]

Similarly we can show $\int_{V_{g_{1}g_{2}g_{3}}}f(v)((\mu_{1}\ast\mu_{2})\ast\mu_{3})(dv)$ also equals to the right hand side the above formula. Therefore the associativity is proved. Next we prove $\pi(g_{1},\mu_{1})\pi(g_{2},\mu_{2})=\pi(g_{1}g_{2},\mu_{1}\ast\mu_{2})$. For any $f\in{\cal S}(R_{-}^{n})$, we have

[TABLE]

[TABLE]

This completes the proof. ∎

Since every element $g\in GL_{n}(F)$ can be written as $g=t^{k}h$ for $h\in M_{n}(R)^{\prime}$ and $k\leq 0$, we need to invert the operator $\pi(tI_{n},\mu)$ to extend the $\pi$-action. A convenient way is to consider the eigenspace of $\pi(tI_{n},\mu)$, and we then are lead to study the elements in $\widehat{M_{n}}(R)^{\prime}$ that commutes with $\pi(tI_{n},\mu)$.

For $u\in GL_{n}(R)$, $uR^{n}=R^{n}$, so $V_{u}=\{0\}$. We denote $\mu_{st}$ for the counting measure on the one point set $V_{u}=\{0\}$. Then for any $f\in\mathcal{S}(R_{-}^{n})$, we have

[TABLE]

For ${\bf k}=(k_{1},\cdots,k_{n})\in{\mathbb{Z}}_{\geq 0}^{n}$, let

[TABLE]

It is easy to see that

[TABLE]

Let $\mu_{st}$ denote the Haar measure induced by the inner product on $V_{t^{\bf k}}$ on which $\{t^{-i_{j}}e_{j}\}$ is an orthonormal basis.

It is easy to show that, for $u,v\in GL_{n}(R)$,

[TABLE]

and

[TABLE]

We also have for $c\in{\mathbb{R}},c\neq 0$,

[TABLE]

More generally, for $u\in GL_{n}({\mathbb{R}}[[t]])$,

[TABLE]

where for $a>0$, $(g,\mu)\in\widehat{M_{n}}(R)^{\prime}$, $a(g,\mu)=(g,a\mu)$ and $a\mu$ is the Haar measure on $V_{g}$ obtained by rescaling $\mu$ by $a$. We prove (2.8) by using the representation ${\cal S}(R_{-}^{n})$, the other identities can be proved similarly.

Proof of (2.8). By (2.7), it suffices to prove for $k=1$. For any $f\in\mathcal{S}(R^{n}_{-})$,

[TABLE]

We will write

[TABLE]

For an eigenvalue $\lambda\in{\mathbb{C}}$, $\lambda\neq 0$, we consider the eigenspace of $\pi_{t}=\pi(tI_{n},\mu_{st})$

[TABLE]

We will prove this space is a representation of a central extension of a subgroup of $GL_{n}(F)$ that is slightly larger than $SL_{n}(F)$, which is given by

[TABLE]

where the leading coefficient refers to the coefficient of the lowest power of $t$. Obviously $GL_{n}(F)_{0}$ is a subgroup of $GL_{n}(F)$.

Lemma 2.2**.**

For $g\in GL_{n}(F)_{0}\cap M_{n}(R)^{\prime}$, $k\in{\mathbb{Z}}_{>0}$, and any Haar measure $\mu$ on $V_{g}$, $(g,\mu)$ and $(t^{k}I_{n},\mu_{st})$ commutes in $\widehat{M_{n}}(R)^{\prime}$, i.e.,

[TABLE]

Proof. Since $\pi_{t^{k}}=\pi_{t}^{k}$, it is enough to prove the case $k=1$. By the Bruhat decomposition (see e.g., [GR])

[TABLE]

we can prove that

[TABLE]

where

[TABLE]

We write $g$ as $g=h_{1}t^{\bf k}h_{2}$ according to the last decomposition, ${\bf k}=(k_{1},\dots,k_{n})$, $t^{\bf k}=h_{1}^{-1}gh_{2}^{-1}$ has entries in $R$, so all $k_{i}\geq 0$. $(g,\mu)=(h_{1},\mu_{st})(t^{\bf k},\mu_{st})(h_{2},\mu_{st})$ up to a scalar, the results follows from (2.7) and (2.9). $\Box$

By Lemma 2.2, the space ${\cal S}(R_{-}^{n})_{\lambda}$ is stable under $\pi(g,\mu)$ for $g\in GL_{n}(F)_{0}\cap M_{n}(R)^{\prime}$. We may extend the $\pi$-action on ${\cal S}(R_{-}^{n})_{\lambda}$ to a central extension of the group $GL_{n}(F)_{0}$. We first define the central extension. Let $\widehat{M_{n}}(R)_{0}^{\prime}$ denote the set of pairs $(g,\mu)\in\widehat{M_{n}}(R)^{\prime}$ with $g\in GL_{n}(F)_{0}\cap M_{n}(R)^{\prime}$, it is a semisubgroup of $\widehat{M_{n}}(R)^{\prime}$. Consider the direct product semigroups

[TABLE]

we define an equivalence relation $\sim$ on $\tilde{G}$ induced by

[TABLE]

It is clear that $\sim$ is compatible with semi-group structure of $\tilde{G}$, therefore the set $\tilde{G}/\sim$ of equivalence classes is a semigroup. We prove it is a group and is a central extension of $GL_{n}(F)_{0}$.

Lemma 2.3**.**

$\tilde{G}/\sim$* is a group and $\tilde{G}/\sim\to GL_{n}(F)_{0}$, $(t^{k}I_{n},(g,\mu))\mapsto t^{k}g$ is a surjective group homomorphism with kernel isomorphic to ${\mathbb{R}}_{>0}$.*

Proof. The only non-trivial part is the the existence of inverse in $\tilde{G}/\sim$. For $\alpha\in G/\sim$ represented by $(t^{k}I_{n},(g,\mu))$, let $l$ large enough so $t^{l}g^{-1}\in M_{n}(R)^{\prime}_{0}$, choose an Haar measure $\nu$ on $V_{t^{l}g^{-1}}$, then

[TABLE]

$(t^{l}I_{n},\mu*\nu)=c(t^{l}I_{n},\mu_{st})$ for some $c>0$, so the right hand side of (2.12) is

[TABLE]

which is invertible in $\tilde{G}/\sim$.∎

We will the denote the group $\tilde{G}/\sim$ in the above lemma by $\widehat{GL_{n}}(F)_{0}$, which is a central extension of ${GL_{n}}(F)_{0}$ by ${\mathbb{R}}_{>0}$:

[TABLE]

And we denote the inverse image of $SL_{n}({\mathbb{R}}((t)))$ by $\widehat{SL_{n}}({\mathbb{R}}((t)))$.

Theorem 2.4**.**

${\cal S}(R^{n}_{-})_{\lambda}$* is a representation of $\widehat{GL_{n}}(F)_{0}$, an element in $\widehat{GL_{n}}(F)_{0}$ represented by $(t^{l},(g,\mu))$ acts as*

[TABLE]

And ${\cal S}(R^{n}_{-})_{\lambda}$ is a representation of $\widehat{SL_{n}}(F)$ by restriction.

Lemma 2.5**.**

On $\mathcal{S}(R^{n}_{-})$, for any nonzero constant $c$, we have

[TABLE]

And $\pi_{c}$ is an isomorphism of the $\widehat{SL_{n}}(F)$-modules

Proof. From (2.8) with $k=1$, we see that

[TABLE]

then using $\pi_{c^{-1}}$, we see the equality of (2.14) holds. It remains to prove that $\pi_{c}\pi(g)=\pi(g)\pi_{c}$ for $g\in\widehat{SL_{n}}(F)$. Use the Bruhat decomposition of $\widehat{SL_{n}}(F)$, it is enough to prove the cases $g\in SL_{n}(R)$ and $g=(t^{l}I_{n},(t^{\bf k},\mu_{st}))$, where

[TABLE]

with $k_{1}+\cdots+k_{n}+ln=0$. The case $g\in SL_{n}(R)$ follows from (2.6). For the later case, let $f\in{\cal S}(R^{n}_{-})_{\lambda}$, from (2.14), $\pi_{c}f\in{\cal S}(R^{n}_{-})_{|c|^{n}\lambda}$, we have, ${\rm dim}\,V_{t^{\bf k}}=k_{1}+\dots+k_{n}=-nl$,

[TABLE]

On the other hand side,

[TABLE]

This proves the lemma. $\Box$

The central extension splits in the subgroup $G$ given in (1.2). It is easy to see that $G$ commutes with $\widehat{SL_{n}}(F)$. For every character $\chi:G\to{\mathbb{C}}^{*}$, the space of $\chi$-coinvariants

[TABLE]

where $N(\chi)$ is the linear span of elements $\pi(g)f-\chi(g)f$, is a representation of $\widehat{SL_{n}}(F)$. Since $\pi_{c}$ commutes with $G$, it induces an isomorphism

[TABLE]

According to the analogy with the classical theta correspondence, $\mathcal{S}(R^{n}_{-})_{\lambda,\chi}$ should be understood as a degenerate principal series representation. The principal series for $\widehat{\mathfrak{s}l}_{2}$ is considered in [FZ] using a different method.

We will write an element $x\in R_{-}^{n}={\mathbb{R}}^{n}[t^{-1}]t^{-1}$ as

[TABLE]

where $x_{-i}\in{\mathbb{R}}^{n}$. We write a function $f$ on $R_{-}^{n}$ as

[TABLE]

accordingly. Sometimes it is more convenient to write each $x_{-i}\in{\mathbb{R}}^{n}$ as $x_{-i}=\sum_{j=1}^{n}x_{-i}^{j}e_{j}$, where $e_{j}$ ($j=1,2,\dots,n$) is the standard basis for ${\mathbb{R}}^{n}$, so

[TABLE]

$f\in\mathcal{S}(R_{-}^{n})$ is expressed as a function of variables $x^{j}_{-i}$:

[TABLE]

3. Maximal Compact Subgroup of $\widehat{SL_{n}}(F)$ and Its Fixed Gaussian Function.

In this section, we give a construction of a “maximal compact subgroup” $K$ of $\widehat{SL_{n}}(F)$ and give a Gaussian function in $\mathcal{S}(R^{n}_{-})_{\lambda}$ that is fixed by $K$.

We first define a “maximal compact subgroup” $O$ of $GL_{n}(F)$ by

[TABLE]

where $g(t^{-1})^{T}$ denotes the transpose of $g(t^{-1})$. Let $d(t)={\rm det}\,g(t)\in{\mathbb{R}}[t,t^{-1}]$. The condition $g(t)g(t^{-1})^{T}=I_{n}$ implies that $d(t)d(t^{-1})=1$, so $d(t)=\pm t^{k}$ for some $k\in{\mathbb{Z}}$. This proves $O\subset GL_{n}(F)_{0}$. We will prove the central extension $\widehat{GL_{n}}(F)_{0}$ splits over $O$.

Our definition of the “orthogonal group” $O$ is justified by the following consideration. We first define an inner product on $\mathbb{R}^{n}[t,t^{-1}]$. For any $u(t),v(t)\in\mathbb{R}^{n}[t,t^{-1}]$, define

[TABLE]

where $(w)_{0}$ denotes the constant term in $w$, for any $w\in\mathbb{R}[t,t^{-1}]$. It is clear that homogeneous pieces ${\mathbb{R}}^{n}t^{j}$ are mutually orthogonal and the basis $\{e_{i}t^{j}\}_{1\leq i\leq n,j\in\mathbb{Z}}$ is orthonormal, where $e_{1},\cdots,e_{n}$ denotes the standard basis for $\mathbb{R}^{n}$. An element $g(t)\in{GL}_{n}(\mathbb{R}[t,t^{-1}])$ preserves the inner product (3.2) iff

[TABLE]

for all $u(t),v(t)\in{\mathbb{R}}^{n}[t,t^{-1}]$ iff $g(t)^{T}g(t^{-1})=I_{n}$. Therefore $O$ is the subgroup in $GL_{n}({\mathbb{R}}[t,t^{-1}])$ that preserves the inner product (3.2).

It is clear that $g\in O$ implies that $t^{k}g\in O$ for every $k\in{\mathbb{Z}}$. So every $g\in O$ can be written as $g=t^{k}g^{\prime}$ with $g^{\prime}\in O\cap M_{n}({\mathbb{R}}[t])$. For any $g\in O\cap M_{n}({\mathbb{R}}[t])$, we identify the quotient space ${V}_{g}$ with $g^{-1}R^{n}\cap R^{n}_{-}\subset R^{n}_{-}$. The inner product on ${\mathbb{R}}^{n}[t,t^{-1}]$ restricts to an inner product on $V_{g}$. Let $\mu_{g}$ denote the Haar measure induced by this inner product in the sense that the volume of parallelotope spanned by an orthonormal basis in $V_{g}$ equals $1$. From (2.4), we can easily check

[TABLE]

For any $g\in O$, let $l$ be large enough so that $t^{l}g\in M_{n}({\mathbb{R}}[t])$, then define $s(g):=(t^{-l},(t^{l}g,\mu_{t^{l}g}))$. $s(g)$ is independent of the choice $l$ by (3.3). Hence $s(O)=\{s(g)\,|\,g\in O\}$ is a subgroup of $\widehat{GL_{n}}(F)_{0}$ that is a lifting of $O\subset{GL}_{n}(F)_{0}$. We define a “maximal compact subgroup” $K$ for $\widehat{SL_{n}}(F)$ as

[TABLE]

which is a lifting of $O\cap SL_{n}(F)$.

Now we want to construct a nontrivial function in $\mathcal{S}(R^{n}_{-})_{\lambda}$ that is fixed by $K$. By simple observation we know for any function $f\in\mathcal{S}({\mathbb{R}}^{n})$ with

[TABLE]

then

[TABLE]

The infinite product makes sense since almost all of $x_{-m}^{j}$ are [math] and $f(0)=1$. We take

[TABLE]

Then

[TABLE]

By calculation, we have

[TABLE]

where

[TABLE]

Fix $\lambda>0$, let $c=\lambda^{-\frac{2}{n}}$, then $\pi_{t}\phi_{c}(x)=\lambda\phi_{c}$.

Proposition 3.1**.**

The function $\phi_{c}\in\mathcal{S}(R^{n}_{-})_{\lambda}$ is fixed by $K$.

Proof. Let $(t^{l},(g,\mu_{g}))\in K$, with $g\in O\cap M_{n}(\mathbb{R}[t])$. According to definition, we have

[TABLE]

where we identify $V_{g}$ with $g^{-1}R^{n}\cap R^{n}_{-}$, and $(x)_{-}:=\pi_{-}(x)$.

Since $g\in O\cap M_{n}(\mathbb{R}[t])$, obviously $g(t)^{-1}=g(t^{-1})^{T}\in M_{n}(\mathbb{R}[t^{-1}])$. This means $g(t)^{-1}x$ is indeed an element in $R^{n}_{-}$, and hence $(g^{-1}x)_{-}=g^{-1}x$. Then since $g$ preserves the $(\cdot,\cdot)$, we have

[TABLE]

But $gy\in R^{n}$ as $y\in g^{-1}R^{n}\cap R^{n}_{-}$. The orthogonality of $x$ and $gy$ implies the right side of the above equation equals

[TABLE]

Thus

[TABLE]

And as $\det t^{l}g=1$, the dimension of $V_{g}$ equals $-ln$. Combining with the fact that $\lambda=c^{-\frac{n}{2}}$, leads to the desired result. $\Box$

4. Formula for action of Affine Lie Algebra $\widehat{\mathfrak{sl}_{n}}$.

In this section, we derive a formula for the action of affine algebra of $\mathfrak{sl}_{n}$ on ${\cal S}(R_{-}^{n})_{\lambda}$ corresponding to the loop group action constructed in Section 2. Let $\mathfrak{g}=\mathfrak{sl}_{n}$ and $\hat{\mathfrak{g}}$ be the corresponding affine Kac-Moody algebra, so

[TABLE]

with Lie bracket given by

[TABLE]

where $(a,b)={\rm tr}\,ab$. By (2.5), $GL_{n}(R)$ acts on ${\cal S}(R_{-}^{n})$, its Lie algebra $gl_{n}\otimes{\mathbb{C}}[t]$ also acts on the same space. We write an function $f\in{\cal S}(R_{-}^{n})$ as (2.17). The action of $gl_{n}\otimes{\mathbb{C}}[t]$ is given by first order differential operator as in the following lemma.

Lemma 4.1**.**

Let $E_{uv}\in gl_{n}$ be the matrix with $(u,v)$-entry $1$ and other entries [math]. For $j\geq 0$, $E_{uv}t^{j}\in gl_{n}\otimes{\mathbb{C}}[t]$, let

[TABLE]

This gives a representation of $gl_{n}\otimes{\mathbb{C}}[t]$ on ${\cal S}(R_{-}^{n})$.

Proof. We first argue that $\pi(E_{uv}t^{j})f\in{\cal S}(R_{-}^{n})$ for $f\in{\cal S}(R_{-}^{n})$. It is enough to prove, for every $m\geq 1$, the restriction of $\pi(E_{uv}t^{j})f$ to ${\mathbb{R}}^{n}t^{-1}+\dots+{\mathbb{R}}^{n}t^{-m}$ is a Schwartz function. This restriction is

[TABLE]

which is in ${\cal S}[{\mathbb{R}}^{n}t^{-1}+\dots+{\mathbb{R}}^{n}t^{-m}]$. It can be proved by a direct calculation that $[\pi(a),\pi(b)]=\pi([a,b])$ for $a,b\in gl_{n}[t]$. $\Box$

The formula (4.1) is derived from the formal calculation

[TABLE]

[TABLE]

[TABLE]

Taking derivative $\frac{d}{d\epsilon}|_{\epsilon=0}$, we derive (4.1).

Theorem 4.2**.**

Let $\pi:\hat{\mathfrak{g}}\to End({\cal S}(R_{-}^{n})_{\lambda})$ be the linear map given as follows: $\pi(at^{j})$ is (4.1) for $a\in\mathfrak{g}$ and $j\geq 0$; for $j>0$, $u\neq v$,

[TABLE]

and

[TABLE]

Then this gives a representation of affine Kac-Moody algebra $\hat{\mathfrak{g}}$ on ${\cal S}(R_{-}^{n})_{\lambda}$ with central charge $1$.

Proof. We have already proved in the proof of Lemma 4.1 that $\pi(at^{j})$ for $j\geq 0$ preserves the space of Schwartz functions. The similar method applies to show that $\pi(at^{-j})$, $j\geq 1$ sends a Schwartz function on $R_{-}^{n}$ to a Schwartz function on $R_{-}^{n}$. Then we need to prove that the operators $\pi(at^{j})$ for $a\in\mathfrak{g}$ and $j\geq 0$ and operators in (4.2), (4.3) preserve the eigenspace ${\cal S}(R_{-}^{n})_{\lambda}$. To this end, we need the relations:

[TABLE]

Using

[TABLE]

one checks the first two relations directly. The third follows from the fact that

[TABLE]

for a Schwartz function $f\in{\cal S}({\mathbb{R}})$. Using (4.4), one can check the operators $\pi(at^{i})$, $j\in{\mathbb{Z}}$, commute with $\pi_{t}$ therefore they preserves the eigenspace ${\cal S}(R_{-}^{n})_{\lambda}$. We verify $\pi_{t}$ commutes with (4.2).

[TABLE]

The other commutation relations can be verified similarly. Next we prove $[\pi(a),\pi(b)]=\pi([a,b])$ for $a,b\in\hat{\mathfrak{g}}$ and the central charge $K$ acts as $1$. The case $a,b\in\mathfrak{sl}_{n}[t]$ has already appeared in Lemma 4.1. We compute the case $a=E_{vu}t$ and $b=E_{uv}t^{-1}$. The other cases are similar. Notice that

[TABLE]

the last “$=$” follows from the fact that $\pi_{t}$ acts on ${\cal S}(R_{-}^{n})_{\lambda}$ as $\lambda$. Now

[TABLE]

[TABLE]

[TABLE]

Combine (4.6), (4.7) and (4.8), we get

[TABLE]

This proves $K=1$. $\Box$

Next we explain how (4.2) is derived from the group action in Section 2. The case (4.3) is similar. To ease the notations, we assume $n=2$ and $E_{uv}=E_{12}$. Notice that

[TABLE]

and $g\in O$. We show $\pi(E_{12}t^{-j})$ given in (4.2) is equal to $\pi(g)\pi(E_{12}t^{j})\pi(g^{-1})$, where $\pi(E_{12}t^{j})$ is given by (4.1).

[TABLE]

To write $\pi(g)$ and $\pi(g^{-1})$ more explicitly, we introduce operators $\pi_{t,1}$ and $\pi_{t,2}$ on ${\cal S}(R_{-}^{2})$.

[TABLE]

[TABLE]

Then

[TABLE]

Notice that the relations $\pi_{t}=\pi_{t,1}\pi_{t,2}=\pi_{t,2}\pi_{t,1}$.

[TABLE]

and

[TABLE]

[TABLE]

We apply relations to move $\pi_{t,2}^{j}$ to the right and $\pi_{t,1}^{j}$ to the left, we find the result is (4.2).

Parallel to the result in the section 2, we will show the function $\phi_{c}$ is killed by a subalgebra of $\hat{\mathfrak{g}}$.

Proposition 4.3**.**

The function $\phi_{c}\in\mathcal{S}(R^{n}_{-})_{\lambda}$ as in (3.4) is killed by the fixed points of Chevalley involution of $\hat{\mathfrak{g}}$.

Proof. The fixed points of Chevalley involution $w$ is ${\rm Span}\{at^{m}-a^{T}t^{-m}\;|\;a\in\mathfrak{sl}_{n},m\in\mathbb{Z}\}$, where $w(at^{m})=-a^{T}t^{-m},\;w(K)=-K$. So suffices to prove $h_{c}$ is killed by ${\rm Span}\{at^{m}-a^{T}t^{-m}\;|\;a\in\mathfrak{sl}_{n},m\in\mathbb{Z}\}$.

We first verify $\pi(E_{uv}t^{m}-E_{vu}t^{-m})\phi_{c}=0$ for $u\neq v$.

[TABLE]

Hence

[TABLE]

For the case $\pi((E_{uu}-E_{vv})(t^{m}-t^{-m}))\phi_{c}=0$, similar calculation shows

[TABLE]

Hence

[TABLE]

∎

5. Relations With Highest Weight Modules

In this section, we show that certain highest weight modules of $\hat{\mathfrak{g}}$ appear in the dual representation of the representation ${\cal S}(R^{n}_{-})_{\lambda}$.

Let $\mathcal{S}(R^{n}_{-})_{\lambda}^{\ast}$ be the full dual space of $\mathcal{S}(R^{n}_{-})_{\lambda}$. It is representation of affine Lie algebra $\hat{\mathfrak{g}}$ under the action

[TABLE]

It is clear that the level of $\mathcal{S}(R^{n}_{-})_{\lambda}^{\ast}$ is $-1$ as $K$ acts as $-1$ by (5.1). This representation is too big to be interesting. However we show that some natural linear functionals are highest weight vectors.

Let $\mathfrak{h}$ be its Cartan subalgebra of $\mathfrak{g}=\mathfrak{sl}_{n}$ consisting of diagonal matrices, $n^{+}$ ($n^{-}$) be the strictly upper (lower) triangular matrices, so we have triangular decomposition $\mathfrak{g}=n^{-}\oplus\mathfrak{h}\oplus n^{+}$. The corresponding triangular decomposition for $\hat{\mathfrak{g}}$ (see [Ka]) is

[TABLE]

with

[TABLE]

We define linear functional $I_{k}$ for $1\leq k\leq n$ as follows, for $f\in\mathcal{S}(R^{n}_{-})_{\lambda}$,

[TABLE]

where $d$ is the Lebesgue measure on $\mathbb{R}$.

We consider $I_{n}$ first. For any $E_{uv}t^{j},\;j\geq 1$,

[TABLE]

by (4.1), which is [math] by (5.2). Hence $\pi^{\ast}(E_{uv}t^{j})I_{n}=0,$ for $j\geq 1$. Similarly by (4.1), we have

[TABLE]

The above formula implies immediately $\pi^{\ast}(E_{uv})I_{n}=0$ for $u\neq v$ since $f$ is a Schwartz function. Also it implies $\pi^{\ast}(E_{uu})I_{n}=-I_{n}$ due to $x_{-1}^{u}\partial_{x_{-1}^{u}}=-[\partial_{x_{-1}^{u}},x_{-1}^{u}]+\partial_{x_{-1}^{u}}x_{-1}^{u}=-1+\partial_{x_{-1}^{u}}x_{-1}^{u}$. Hence $\pi^{\ast}(E_{uu}-E_{vv})$ also kills $I_{n}$. Thus we have shown the dual action of any element in $\mathfrak{sl}_{n}(R)=\mathfrak{sl}_{n}(\mathbb{R}[[t]])$ kills $I_{n}$. Moreover, the central charge $K$ acts as $-1$ as mentioned earlier.

Similarly one can show $\pi^{\ast}(E_{uv}t^{j})$ for $j\geq 1$ kills $I_{k}$. Now we consider $\pi^{*}(E_{uv})$ for $u\leq v$. If $v>k$, we see that $\pi^{\ast}(E_{uv})I_{k}=0$ immediately from the definition of $I_{k}$. For $u,v\leq k$, similar to (5.4), we derive that $\pi^{\ast}(E_{uv})$ kills $I_{k}$ when $u\neq v$ and acts as $-1$ when $u=v$. To conclude, we obtain the following theorem

Theorem 5.1**.**

For $k=1,\cdots,n$, $I_{k}\in\mathcal{S}(R^{n}_{-})_{\lambda}^{\ast}$ is a highest weight vector with highest weight $\Lambda_{k}$ given as follows:

[TABLE]

∎

6. Whittaker Functionals

Let’s denote the Lie algebra of linear map on $n\times n$ matrix space $M_{n}$ by $gl_{n^{2}}$ and the group of invertible linear transformations on $M_{n}$ by $GL_{n^{2}}$. We have the Lie algebra embedding $\mathfrak{sl}_{n}\oplus\mathfrak{sl}_{n}\to gl_{n^{2}}$ by the action

[TABLE]

and the corresponding group embedding $SL_{n}\oplus SL_{n}\to GL_{n^{2}}$ is given by the action

[TABLE]

As in Section 2 and 4, we know ${\cal S}(M_{n}({\mathbb{R}})[t^{-1}]t^{-1})_{\lambda}$ is a representation of $\widehat{SL_{n^{2}}}({\mathbb{R}})$ and affine Lie algebra $\widehat{\mathfrak{sl}_{n}}$, therefore is a representation of $\widehat{SL_{n}}({\mathbb{R}})\times\widehat{SL_{n}}({\mathbb{R}})$ and Lie algebra $\widehat{\mathfrak{sl}_{n}}\oplus\widehat{\mathfrak{sl}_{n}}$ by restriction, its level is $n$. The dual space ${\cal S}(M_{n}({\mathbb{R}})[t^{-1}]t^{-1})_{\lambda}^{*}$, as a representation of $\widehat{\mathfrak{sl}_{n}}\oplus\widehat{\mathfrak{sl}_{n}}$, has level $-n$, which is the critical level as the dual Coxeter number of $\mathfrak{sl}_{n}$ is $n$. We will construct Whittaker functionals ${\cal S}(M_{n}({\mathbb{R}})[t^{-1}]t^{-1})_{\lambda}^{*}$. The Whittaker functionals for $\widehat{sl}_{2}$ are studied abstractly in [ALZ], our construction provides a concrete realization.

We denote the action of the first (second) copy of $\widehat{\mathfrak{sl}_{n}}$ in $\widehat{\mathfrak{sl}_{n}}\oplus\widehat{\mathfrak{sl}_{n}}$ by $\pi_{1}(a)$ ($\pi_{2}(a)$) and the similar notations $\pi_{1}(g)$ and $\pi_{2}(g)$ will be used for loop groups. Same convention will be used for the corresponding dual action $\pi_{1}^{\ast}$ and $\pi_{2}^{\ast}$ on the dual space ${\cal S}(M_{n}({\mathbb{R}})[t^{-1}]t^{-1})_{\lambda}^{*}$.

As in Section 2, we will write an element $x\in M_{n}(\mathbb{R})[t^{-1}]t^{-1}$ as

[TABLE]

where $x_{-i}\in M_{n}(\mathbb{R})$. We write a function $f$ on $M_{n}(\mathbb{R})[t^{-1}]t^{-1}$ as

[TABLE]

or as a function of variables $x_{-i}^{u,v}$:

[TABLE]

where $x_{-i}^{u,v}$ denotes $(u,v)$-entry of $x_{-i}$.

By (4.1) (4.2) (4.3), we have the following formulas

[TABLE]

and for $j>0,u\neq v$,

[TABLE]

and

[TABLE]

[TABLE]

and for $j>0,u\neq v$,

[TABLE]

and

[TABLE]

In this setting, take $\phi_{c}=e^{-\pi c\sum_{1\leq u,v\leq n}\sum_{k=1}^{\infty}(x^{u,v}_{-k})^{2}}$ and $c=\lambda^{-\frac{2}{n^{2}}}$. We can easily show that $\phi_{c}$ is fixed by the maximal compact subgroup $K\times K$ and killed by its Lie algebra as in Section 3.

Next we introduce Whittaker functionals in ${\cal S}(M_{n}({\mathbb{R}})[t^{-1}]t^{-1})_{\lambda}^{*}$. Our formula is motivated by the following finite dimensional case. For simplicity we consider the case $GL_{3}({\mathbb{R}})\times GL_{3}({\mathbb{R}})$. It acts on the Schwartz space ${\cal S}(M_{3}({\mathbb{R}}))$ as follows,

[TABLE]

We still denote the action of the first (second) copy of $GL_{3}({\mathbb{R}})$ in $GL_{3}({\mathbb{R}})\times GL_{3}({\mathbb{R}})$ by $\pi_{1}(g)$ ($\pi_{2}(g)$).

Let $B$ be the standard Borel subgroup of $GL_{3}({\mathbb{R}})$, i.e., $B$ consists of upper triangular matrices. Then $B^{T}$, the lower triangular matrices, is an opposite Borel subgroup. $B_{2}=B\times B^{T}$ is a Borel subgroup of $GL_{3}({\mathbb{R}})\times GL_{3}({\mathbb{R}})$. For $c=(c_{1},c_{2})\in{\mathbb{R}}^{2}$, we define a linear functional $\Phi_{c}:\cal S(M_{3}(\mathbb{R}))\to{\mathbb{C}}$ by

[TABLE]

Let $\chi_{c}$ be the character of the unipotent radical $U$ of $B$ defined by

[TABLE]

For such $u\in U$, we have

[TABLE]

We check that

[TABLE]

where $\pi_{1}^{\ast}$ is the dual action of $\pi_{1}$ on the full dual space of $\cal S(M_{3}(\mathbb{R}))$. Indeed, by applying the functional $\Phi_{c}$ to $\pi_{1}(u)^{-1}f$ and using (6.11), the left side equals

[TABLE]

After changing variables, the above formula will be turned into

[TABLE]

which is indeed the right side of (6.12).

Thus the functional $\Phi_{c}$ in (6.10) is a Whittaker functional for the first copy of $GL_{3}(\mathbb{R})$. Similarly we will also give the Whittaker functional for the second copy of $GL_{3}(\mathbb{R})$ as follows: For $d=(d_{1},d_{2})=\mathbb{R}^{\ast}$, define a linear functional $\Phi_{d}^{\prime}:\cal S(M_{3}(\mathbb{R}))\to\mathbb{C}$ by

[TABLE]

Let $\chi_{d}$ be a character of the unipotent radical $U^{T}$ in $B^{T}$ defined by

[TABLE]

In this case we have

[TABLE]

where $\pi_{2}^{\ast}$ is the dual action of $\pi_{2}$ on the full dual space of $\cal S(M_{3}(\mathbb{R}))$.

We consider the loop group case $\widehat{SL_{n}}(F)\times\widehat{SL_{n}}(F)$. It acts on the space $\cal S(M_{n}(\mathbb{R})[t^{-1}]t^{-1})_{\lambda}$ with the action explained in the beginning of section 6. Let $\hat{B}$ be the Borel subgroup consisting of elements $b\in\widehat{SL_{n}}(R)$ such that $b(0)$ is upper triangular. Then $\hat{B}^{T}$, is an opposite Borel subgroup. And $\hat{B}\times\hat{B}^{T}$ is a Borel subgroup of $\widehat{SL_{n}}(F)\times\widehat{SL_{n}}(F)$.

For $c=(c_{1},c_{2},\cdots,c_{n})\in\mathbb{R}^{n}$, we define a linear functional $\Psi_{c}:\cal S(M_{n}(\mathbb{R}[t^{-1}]t^{-1}))\to\mathbb{C}$ by

[TABLE]

Let $\hat{\chi}_{c}$ be a character on the unipotent subgroup of $B$ defined as follows: For any $u\in B$ such that $u(0)$ is upper triangular unipotent, that is

[TABLE]

then define

[TABLE]

where $u_{1},\cdots,u_{n-1}$ are superdiagonal elements of $u\in B$, and $v_{n}$ is the $(n,1)$-entry of $u^{\prime}(0)$.

Now we show that $\Psi_{c}$ is indeed a Whittaker functional of $\widehat{SL_{n}}(F)$ under the group action $\pi_{1}$ on $\cal S(M_{n}(\mathbb{R})[t^{-1}]t^{-1})_{\lambda}$ in the sense that

[TABLE]

Indeed, by $(\pi_{1}^{\ast}(u)\Psi_{c},f)=(\Psi_{c},\pi_{1}(u)^{-1}f)$, the left side equals

[TABLE]

Since the Jacobi matrix is upper triangular with all of diagonal entries equals one, hence the Jacobi determinant equals $1$. And by changing the variables, the above formula will be turned into

[TABLE]

which is exactly the right side of (6.16).

Therefore the functional (6) gives a Whittaker functional for the first copy of $\widehat{SL_{n}}(F)$. Similarly we will also give the Whittaker functional for the second copy of $\widehat{SL_{n}}(F)$ as follows: For $d=(d_{1},\cdots,d_{n})\in\mathbb{R}^{n}$, define a linear functional $\Psi_{d}^{\prime}:\cal S(M_{n}(\mathbb{R})[t^{-1}]t^{-1})_{\lambda}\to\mathbb{C}$ by

[TABLE]

Let $\hat{\chi}_{d}$ be a character of the unipotent subgroup in $B^{T}$ defined as follows: For any $v\in B^{T}$ such that $v(0)$ is lower triangular unipotent, that is

[TABLE]

then define

[TABLE]

where $v_{1},\cdots,v_{n-1}$ are subdiagonal elements of $v\in B^{T}$, and $z_{n}$ is the $(1,n)$-entry of $v^{\prime}(0)$. Similar to the computation before, we have

[TABLE]