UDC 519.46

MATHEMATICS

I. N. BERNSTEIN, Corresponding Member of the Academy of Sciences of the USSR I. M. GELFAND,
S. I. GELFAND

DIFFERENTIAL OPERATORS ON THE BASIC AFFINE SPACE

Let \(G\) be a connected semisimple algebraic Lie group of rank \(r\) over an algebraically closed field \(K\) of characteristic \(0\), let \(B\) be its Borel subgroup, \(N\) the unipotent radical of \(B\), and \(H\) a Cartan subgroup contained in \(B\).

A fundamental role in representation theory is played by the space
\(A=N\backslash G\)—the basic affine space of the group \(G\). \(A\) is an algebraic variety. The purpose of this paper is to study the space of differential operators with regular coefficients on \(A\). For the precise definition of a regular differential operator see \((^1)\).

To each function \(f(g)\) on the group \(G\) there will be constructed a set of regular differential operators on \(A\). In addition, we shall show that in this way all differential operators on \(A\) can be obtained. For the construction an operation \(\pi_*\) is introduced, mapping functions on \(G\) to functions on \(A\), which, in our opinion, is of independent interest. This operation is the algebraic analogue of the operation of averaging over a subgroup (in the present case, a unipotent one). Remarkably, \(\pi_*\) takes the operation of multiplication by a function \(f(g)\) into an “almost” differential operator on \(A\). Precise formulations are given in Theorems 1 and 2.

1. Denote by \(\mathcal E(G)\) and \(\mathcal E(A)\) the spaces of regular functions on the algebraic varieties \(G\) and \(A\). Define representations \(R^G\) and \(R^A\) of the group \(G\) in these spaces by right translations

\[ (R^G_{g_0} f)(g)=f(gg_0), \qquad (R^A_{g_0}\varphi)(x)=\varphi(xg_0), \qquad g,g_0\in G,\; x\in A. \]

Then every irreducible invariant subspace in these spaces is finite-dimensional. We can also introduce the representations \(L^G\) of the group \(G\) in \(\mathcal E(G)\) and \(L^A\) of the group \(H\) in \(\mathcal E(A)\) by left translations

\[ (L^G_{g_0} f)(g)=f(g_0^{-1}g), \qquad (L^A_h\varphi)(x)=\varphi(h^{-1}x), \qquad g,g_0\in G,\; h\in H,\; x\in A; \]

the element \(h^{-1}x\) is defined, since \(H\) normalizes \(N\).

Let \(\mathfrak G\) be the Lie algebra of the group \(G\), \(\mathfrak h\) the Cartan subalgebra corresponding to \(H\), and \(\mathfrak N\) the Lie algebra of the group \(N\). Let \(\alpha_i\) \((1\le i\le k)\) be all positive roots of \(\mathfrak h\) in \(\mathfrak G\), \(E_i^{+}\in \mathfrak N\) the corresponding root vectors, and \(E_i\) the root vectors with weights \(\alpha_i\).

Differentiating the representations \(L^G\) and \(L^A\), we may regard elements of \(\mathfrak G\) as right-invariant differential operators of first order on \(G\), and elements of \(\mathfrak h\) as differential operators of first order on \(A\).

1. The operation \(\pi_*\). Denote by \(\pi:G\to A=N\backslash G\) the natural projection and by \(\pi^*:\mathcal E(A)\to \mathcal E(G)\) the mapping given by the formula \((\pi^*\varphi)(g)=\varphi(\pi g)\). Then it is clear that \(R^G_g\pi^*=\pi^*R^A_g\) and \(L_h\pi^*=\pi^*L^A_h\). Note, moreover, that \(\pi^*\) is a monomorphism and \(f\in \operatorname{Im}\pi^*\) if and only if \(E_i f=0\) for all \(i\).

Lemma–definition. There exists a unique mapping
\(\pi_*:\mathscr E(G)\to \mathscr E(A)\) such that

1) \(\pi_* R_g^G = R_g^A \pi_*,\quad \pi_* L_h^G = L_h^A \pi_*\) for all \(g\in G,\ h\in H\);

2) \(\pi_*\pi^*\varphi=\varphi\) for all \(\varphi\in \mathscr E(A)\).

Proof. We shall prove that \(\pi_* f\) is uniquely determined by conditions 1) and 2). Every regular function on \(G\) lies in a finite-dimensional irreducible subspace invariant with respect to \(L^G\). Therefore it is enough to consider the case when \(f\) lies in an irreducible subspace \(V\) invariant with respect to \(L^G\) and is a weight function of weight \(\chi\) with respect to the restriction of \(L^G\) to \(H\). If \(\chi\) is the highest weight of the given irreducible representation, then \(f\in \operatorname{Im}\pi^*\), i.e. \(f=\pi^*\varphi\), \(\varphi\in \mathscr E(A)\), and, in view of 2), \(\pi_*f=\varphi\).

Now let \(\chi\) not be the highest weight. Denote by \(f_0\) the vector of highest weight in \(V\) and by \(\chi_0\) this highest weight. Then \(f\) and \(f_0\) are transformed under the action of \(R^G\) according to one and the same irreducible representation of \(G\). From 1) and from the fact that each irreducible representation of \(G\) occurs in \(\mathscr E(A)\) only once ((1), Lemma 4.1), it follows that \(\pi_*f\) and \(\pi_*f_0\) lie in one and the same irreducible subspace invariant with respect to \(R^G\). But then the weights of \(\pi_*f\) and \(\pi_*f_0\) with respect to \(L^A\) coincide ((1), Lemma 4.2). From 1) it follows that the weight of \(\pi_*f\) is equal to \(\chi\), while the weight of \(\pi_*f_0\) is equal to \(\chi_0\ne \chi\). Since \(\pi_*f_0\ne 0\), we have \(\pi_*f=0\).

This proof immediately also gives the construction of \(\pi_*\). Namely, if \(f\) is a weight vector not of highest weight, lying in an irreducible subspace invariant with respect to \(L^G\), then we set \(\pi_*f=0\). If, however, \(f\) is a vector of highest weight, then \(f=\pi\varphi\), and we set \(\pi_*f=\varphi\). The lemma is proved.

From the construction of \(\pi_*\) it is clear that \(\pi_*(\hat E_i f)=0\) for any function \(f\in \mathscr E(G)\).

3. Construction with the aid of \(\pi_*\) of differential operators on \(A\). Let \(Wu\) be the enveloping algebra of the Lie algebra \(\mathfrak h\). To each element \(w\in Wu\) let us associate right-invariant differential operators in the spaces \(\mathscr E(G)\) and \(\mathscr E(A)\), which we shall denote by the same letter \(w\). The equalities
\[ w\pi^*=\pi^*w,\qquad \pi_*w=w\pi_* \]
hold.

Let \(f\in \mathscr E(G)\). Define in the space \(\mathscr E(A)\) the operator \(\bar f\) by the formula
\(\bar f(\varphi)=\pi_*(f\cdot \pi^*\varphi),\ \varphi\in \mathscr E(A)\).

Theorem 1. There exists a nonzero element \(w\in Wu\) such that \(w\bar f\) is a regular differential operator on \(A\).

Proof. We shall call a chain a differential operator \(D\) on \(G\) of the form \(E_{i_1}E_{i_2}\ldots E_{i_s}\), and the weight of such a chain will be the weight \(a_{i_1}+\cdots+a_{i_s}\). Denote by \(\Xi\) the set of all chains \(D\) such that \(Df\ne 0\), and by \(\Xi_0\) the set of their weights. Obviously, \(\Xi\) and \(\Xi_0\) are finite sets. For any function \(\varphi\in \mathscr E(A)\) and chain \(D\),
\(D(f\cdot \pi^*\varphi)=Df\cdot \pi^*\varphi\), since \(E_i\pi^*\varphi=0\). Therefore, if \(D\notin \Xi\), then \(D(f\pi^*\varphi)=0\). Denote by \(U\) the subspace in \(\mathscr E(G)\) consisting of all functions \(u\) for which \(D(u)=0\) for any chain \(D\notin \Xi\).

Lemma. There exists a regular differential operator \(T\) on \(G\) and an element \(w\in Wu\) such that for all \(u\in U\) the equality
\[ Tu=w\pi^*\pi_*u \]
holds.

The theorem follows immediately from the lemma, since for any function \(\varphi\in \mathscr E(A)\), \(f\cdot \pi^*\varphi\in U\), and hence
\[ T(f\pi^*\varphi)=w\pi^*\pi_*(f\cdot \pi^*\varphi)=\pi^*w\bar f(\varphi), \]
i.e. \(T\circ f\circ \pi^*=\pi^*\circ w\circ \bar f\). From this equality it is clear that the differential operator \(T\circ f\) preserves \(\pi^*\mathscr E(A)\subset \mathscr E(G)\), and therefore \(w\circ \bar f\) is a differential operator on \(A\).

Proof of the lemma. Let \(H_1,\ldots,H_r\) be a basis in \(\mathfrak h\). The elements of \(Wu\) are polynomials in the \(H_i\), and we shall regard them as

polynomial functions on \(\mathfrak h^*\). Let \(\Delta\) be the second-order Laplace operator on \(G\) (constructed from the Killing form). Then there exists an element \(P \in Wu\) such that for any highest-weight vector \(\Psi \in \mathcal E(G)\) the equality \(\Delta \Psi = P\Psi\) holds, or, equivalently, \(\Delta \Psi = P(\chi_0)\Psi\), where \(\chi_0\) is the weight of the vector \(\Psi\).

Let \(B(H)\) be the restriction of the Killing form of the algebra \(\mathfrak G\) to \(\mathfrak h\), and let \(Q(\chi)\) be the dual quadratic form on \(\mathfrak h^*\). From the results of Harish-Chandra \(\left({}^{2}\right)\) it follows that \(P(\chi)=Q(\chi+\rho)-Q(\rho)\), where \(\rho\) is the half-sum of the positive roots.

For any weight \(\beta\), denote by \(P_\beta\) and \(w_\beta\) the elements of \(Wu\) corresponding to the polynomial functions
\[ P_\beta(\chi)=P(\chi+\beta) \]
and
\[ w_\beta(\chi)=2\langle \beta,\chi+\rho\rangle+\langle\beta,\beta\rangle . \]
(\(\langle,\rangle\) is the scalar product on \(\mathfrak h^*\) corresponding to the quadratic form \(Q\).) Let
\[ T=\prod_\beta (P_\beta-\Delta),\qquad w=\prod_\beta w_\beta, \]
where \(\beta\) runs over \(\Xi_0\setminus 0\).

We shall show that for all \(u\in U\) the equality \(Tu=w\pi^*\pi_*u\) holds.

It is clear that it suffices to verify this equality when \(u\) is a weight vector lying in an irreducible \(L^G\)-invariant subspace \(V\). Let \(u_0\) be a highest-weight vector in \(V\), and let \(\chi\) and \(\chi_0\) be the weights of \(u\) and \(u_0\) relative to \(\mathfrak h\). From the uniqueness of the highest-weight vector it follows that \(u_0=cDu\), where \(D\) is some chain, \(c\in K\), and therefore \(\chi_0-\chi\in \Xi_0\).

1st case. \(\chi\ne \chi_0\). The restriction of \(\Delta\) to \(V\) is multiplication by \(P(\chi_0)\). Therefore
\[ (P_\beta-\Delta)u=(P_\beta(\chi)-P(\chi_0))u=(P(\chi+\beta)-P(\chi_0))u=0, \]
if \(\beta=\chi_0-\chi\in\Xi_0\setminus 0\). Hence \(Tu=0\). Since in case 1 \(\pi_*u=0\), also \(w\pi^*\pi_*u=0\), and hence \(Tu=w\pi^*\pi_*u\).

2nd case. \(\chi=\chi_0\). Then
\[ \begin{aligned} (P_\beta-\Delta)u &=(P(\chi_0+\beta)-P(\chi_0))u \\ &=(Q(\chi_0+\beta+\rho)-Q(\chi_0-\beta))u \\ &=(2\langle\chi_0+\rho,\beta\rangle+\langle\beta,\beta\rangle)u =w_\beta(\chi_0)u . \end{aligned} \]
Thus,
\[ Tu=\left(\prod (P_\beta-\Delta)\right)u=\prod w_\beta(\chi_0)u=wu=w\pi^*\pi_*u . \]
The proof of the lemma, and therefore also of Theorem 1, is complete.

It can be shown that the order of the operator \(w\bar f\) is equal to \(\nu(\Xi_0)-1\), i.e., to the order of \(w\). (\(\nu(\Xi_0)\) is the number of elements of \(\Xi_0\).)

Theorem 2. Every regular differential operator \(\mathcal D\) on \(A\) can be represented in the form
\[ \mathcal D=\sum w_j\bar f_j,\qquad \text{where } w_j\in Wu,\quad f_j\in\mathcal E(G). \]

Proof. As shown in \(\left({}^{1}\right), \S 8\), a regular differential operator \(\mathcal D\) on \(A\) extends to a regular differential operator \(\widetilde{\mathcal D}\) on \(G\) (i.e., \(\widetilde{\mathcal D}\pi^*\varphi=\pi^*\mathcal D\varphi\) for all \(\varphi\in\mathcal E(A)\)). The operator \(\widetilde{\mathcal D}\), like every differential operator on \(G\), can be represented in the form
\[ \widetilde{\mathcal D}=\sum s_j(g)P_j, \]
where \(P_j\) are elements of the enveloping algebra \(\mathfrak G\), and \(s_j\in\mathcal E(G)\). By simple transformations \(\widetilde{\mathcal D}\) can be brought to the form
\[ \widetilde{\mathcal D}=\sum w_j f_j+\sum A_iE_i+\sum \hat E_iB_i, \]
where \(A_i\) and \(B_i\) are some differential operators on \(G\), \(w_j\in Wu\), and \(f_j\in\mathcal E(G)\).

Since \(E_i\pi^*\varphi=0\) and \(\pi_*\hat E_i f=0\), we have
\[ \mathcal D\varphi=\pi_*\pi^*\mathcal D\varphi=\pi_*\widetilde{\mathcal D}\pi^*\varphi =\sum w_j\pi_*(f_j\pi^*\varphi)=\sum w_j\bar f_j(\varphi), \]
which proves Theorem 2.

Let \(L\) be the field of fractions of the ring \(Wu\), and let \(R\) be the ring of regular differential operators on \(A\). Then the operation \(f\mapsto\bar f\) extends to the mapping
\[ \Theta:\mathcal E(G)\otimes_K L\to R\otimes_{Wu}L, \]
\(\Theta\) commutes with the action of \(R_g\) and \(L_h\).

Theorem 3. \(\Theta\) is an isomorphism of \(L\)-modules \(\mathcal{E}(G)\underset{K}{\otimes}L\) and \(R\underset{W_u}{\otimes}L\).

Proof. As follows from (1), the dimension, over \(L\), of the space of vectors transforming according to the given irreducible representation of the group \(G\) under the representations \(R^G\) and \(R^A\) in both spaces coincides. By Theorem 2, \(\Theta\) is an epimorphism; hence \(\Theta\) is an isomorphism.

Moscow State University
named after M. V. Lomonosov

Received
29 VI 1970

REFERENCES

\(^{1}\) I. M. Gelfand, A. A. Kirillov, Functional Analysis and Its Applications, 3, no. 1 (1969).
\(^{2}\) Harish-Chandra, Trans. Am. Math. Soc., 70, 28 (1951).