UDC 512.932

MATHEMATICS

A. N. ANDRIANOV

RATIONALITY OF MULTIPLE HECKE SERIES OF THE FULL LINEAR GROUP AND SHIMURA’S HYPOTHESIS ON HECKE SERIES OF THE SYMPLECTIC GROUP

(Presented by Academician Yu. V. Linnik on 5 VIII 1968)

1. Let \(k\) be a field of \(p\)-adic numbers (a finite extension of the field \(\mathbf Q_p\)); \(D\) a central division algebra of finite rank over \(k\); \(\mathfrak O\) a maximal order in \(D\); \(\mathfrak P=(\Pi)\) the maximal ideal in \(\mathfrak O\). Put \(G=GL(n,D)\), the full linear group of matrices of order \(n\) over \(D\);
\(I(G)=\{(g_{ij})\in G,\ g_{ij}\in\mathfrak O\}\); \(U=\{u\in G,\ u^{\pm1}\in I(G)\}\). With respect to the natural topology induced by the topology of \(D\), \(G\) is a locally compact group; \(U\) is an open and compact subgroup of \(G\). Denote by \(L(G,U)\) (respectively \(L(G,U)_{\mathbf Z}\)) the algebras of all continuous functions \(f\) on \(G\) with values in \(\mathbf C\) (respectively in \(\mathbf Z\)) which, for all \(u,u'\in U,\ g\in G\), satisfy the condition
\[ f(ugu')=f(g). \]
Multiplication in \(L(G,U)\) \(\bigl(L(G,U)_{\mathbf Z}\bigr)\) is defined as convolution
\[ (f*\varphi)(g)=\int_G f(gh^{-1})\varphi(h)\,dh \qquad (g\in G), \]
where \(dh\) is the bi-invariant Haar measure on \(G\); \(\int_U dh=1\).

The algebras \(L(G,U)\) and \(L(G,U)_{\mathbf Z}\), called the Hecke algebras of the group \(G\), play an important role in the theory of automorphic forms and zeta-functions connected with the full linear group. The structure of the Hecke algebras of the group \(G\) was determined by Satake \((^1)\). Let us formulate his result.

Let
\[ \Pi_{i,n}=\operatorname{diag}(\underbrace{1,\ldots,1}_{i},\underbrace{\Pi,\ldots,\Pi}_{n-i}),\qquad i=0,1,\ldots,n-1; \]
for \(a\in G\), denote by \(\chi(a)=\chi(a)(g)\) the characteristic function of the double class \(UaU\). Then \(\bigl((^1),\S 8\bigr)\)
\[ L(G,U)=\mathbf C[\chi(\Pi_{1,n}),\ldots,\chi(\Pi_{n-1,n}),\chi(\Pi_{0,n}^{\pm1})], \]
\[ L(G,U)_{\mathbf Z}=\mathbf Z[\chi(\Pi_{1,n}),\ldots,\chi(\Pi_{n-1,n}),\chi(\Pi_{0,n}^{\pm1})], \]
and the elements \(\chi(\Pi_{i,n})\), \(i=0,1,\ldots,n-1\), are algebraically independent over \(\mathbf C\).

One of the basic problems in the theory of Hecke algebras is to find an explicit expression for an arbitrary \(f\in L(G,U)\) \(\bigl(L(G,U)_{\mathbf Z}\bigr)\) in terms of the generators \(\chi(\Pi_{i,n})\). The theory of elementary divisors for the group \(G\) \((^1)\) allows one to restrict oneself to considering functions \(f\) of the form \(\chi(\Pi^{(r_1,\ldots,r_n)})\), where
\[ \Pi^{(r_1,\ldots,r_n)}=\operatorname{diag}(\Pi^{r_1},\ldots,\Pi^{r_n}),\qquad r_i\in\mathbf Z,\quad 0\le r_1\le r_2\le\cdots\le r_n. \]
It is clear that we can obtain any information of the required kind if we possess sufficiently complete information about the so-called multiple Hecke series of the group \(G\), i.e. the formal series
\[ Z_G(t_1,\ldots,t_n)= \sum_{0\le r_1\le\cdots\le r_n} \chi\bigl(\Pi^{(r_1,\ldots,r_n)}\bigr)t_1^{r_1}\cdots t_n^{r_n}. \tag{1} \]

We have proved the following general fact about the series (1):

Theorem 1. The series (1) is a rational function of \(t_1,\ldots,t_n\) with coefficients from \(L(G,U)_{\mathbf Z}\).

Previously, the rationality of the series (1) was known only in the case when
\(t_1=t_2=\cdots=t_n\) \((^2)\).

1. Theorem 1 admits a reformulation in terms of multiple zeta-functions of the group \(G\). Recall \((^1), \S 5\), that a complex-valued continuous function \(\omega\) on \(G\) is called a zonal spherical function on \(G\) (with respect to \(U\)) if the following conditions are satisfied:
   1) \(\omega(ugu')=\omega(g)\) for all \(g\in G,\ u,u'\in U\);
   2) \(\omega(1)=1\);
   3) for every \(\varphi\in L(G,U)\), \(\omega\) is an eigenfunction of the integral operator defined by \(\varphi\), i.e. \(\varphi*\omega=\lambda_\varphi\omega\), where \(\lambda_\varphi\in\mathbf C\). Denote by \(\Omega(G,U)\) the set of all zonal spherical functions on \(G\) with respect to \(U\). Put, for \(i=1,\ldots,n\) and \(g\in G\),

\[ V_i(g)=(N\mathfrak p)^{-r_i}, \]

where \(UgU=U\Pi^{(r_1,\ldots,r_i,\ldots,r_n)}U,\ r_1\leq r_2\leq\cdots r_n\), and \(N\mathfrak p\) is the number of elements of the residue field \(\mathfrak o/\mathfrak p\). We shall call the multiple zeta-function of the group \(G\) with character \(\omega\in\Omega(G,U)\) the integral

\[ \xi_G(s_1,\ldots,s_n;\omega) = \int_{I(G)}\omega(g^{-1})\prod_{i=1}^{n}V_i(g)^{s_i}\,dg, \tag{2} \]

where \(s_i\) \((i=1,\ldots,n)\) are complex variables. It is easy to see that the integral (2) converges absolutely in some domain of the form \(\min_i \operatorname{Re}s_i>N_\omega\).

Theorem 2. For any \(\omega\in\Omega(G,U)\), the function \(\xi_G(s_1,\ldots,s_n;\omega)\) extends meromorphically in all variables \(s_i\) and is a rational function of the variables \(t_i=(N\mathfrak p)^{-s_i}\) \((i=1,\ldots,n)\).

Previously, a special case of this theorem was known for \(s_1=s_2=\cdots=s_n\) \((^2)\).

1. For the field of \(p\)-adic numbers \(k\), denote by \(\mathfrak o\) and \(\mathfrak p=(\pi)\) the ring of integers of \(k\) and the maximal ideal of this ring, respectively. By \(|\ |\) we shall denote the normalized norm in \(k\), i.e. \(|\xi|=q^{-\operatorname{ord}_{\mathfrak p}\xi}\), where \(\xi\in k\), and \(q\) is the number of elements of the residue class field \(\mathfrak o/\mathfrak p\). Let \(J_n=\begin{pmatrix}0&1_n\\-1_n&0\end{pmatrix}\), where \(1_n\) is the identity matrix of order \(n\). Put

\[ S=\operatorname{Sp}(n,k)=\{g\in GL(2n,k);\ {}^tgJ_ng=r(g)J_n,\ r(g)\in k\} \]

—the symplectic group of genus \(n\) over \(k\); \(I(S)=S\cap I(GL(2n,k))\), \(V=\{v\in S;\ v,v^{-1}\in I(S)\}\). Then, with respect to the natural topology, \(S\) is a locally compact group and \(V\) is an open and compact subgroup. Analogously to § 1, for the pair \((S,V)\) one can define the Hecke algebras \(L(S,V)\) and \(L(S,V)_Z\), which are again algebras of polynomials over \(\mathbf C\) and \(\mathbf Z\), respectively, with explicitly written generators \((^1,^3)\). Denote by \(T_n=T_n(g)\) the characteristic function of the set \(X_n=(x\in I(S),\ \operatorname{ord}_{\mathfrak p} r(x)=n)\). Clearly, \(T_n\in L(S,V)_Z\). The Hecke series of the group \(S\) is the formal series

\[ Z_S(t)=\sum_{n=0}^{\infty}T_nt^n. \tag{3} \]

Shimura \((^3)\) conjectured that the series \(Z_S(t)\) is a rational function of \(t\). Following \((^4)\), one can show that Shimura’s conjecture is a consequence of Theorem 2. Thus one has

Theorem 3. The Hecke series (3) of the group \(\operatorname{Sp}(n,k)\) is a rational function of \(t\) with coefficients in \(L(S,V)_Z\).

Previously, the rationality of the series (3) was known for the cases \(n=1\) \((^5)\); \(n=2\) \((^1,^4)\); \(n=3\) \((^4)\). In these cases the series \(Z_S(t)\) was computed explicitly.

1. Analogously to § 2, Theorem 3 admits a reformulation in terms of zeta-functions of the group \(S\). Let \(\omega\in\Omega(S,V)\) (the definition is analogous to that given in § 2 for \(\Omega(G,U)\); see \((^1), \S 5\)). The integral

\[ \xi_S(s;\omega)=\int_{I(S)}\omega(g^{-1})|r(g)|^s\,dg, \tag{4} \]

where \(dg\) is the Haar measure on \(S\); \(s\) is a complex variable, is called the zeta-function of the group \(S\) with character \(\omega\). It is easy to see that the integral (4) converges absolutely in some domain of the form \(\operatorname{Re} s > N_\omega\).

Theorem 4. The function \(\zeta_S(s;\omega)\) extends meromorphically to the entire \(s\)-plane and is a rational function of \(t=q^{-s}\).

1. Concerning the proof of Theorems 1–4, we note that Theorems 1, 3, 4 are derived from Theorem 2. To prove Theorem 2, we construct special decompositions for the factor-space \(U \setminus GL(n,D)\), reminiscent of the Bruhat decomposition for \(GL(n,D)\), which make it possible to parametrize the left classes \(Ua\) contained in a given double class \(UgU\) \((a,g\in G)\) by points on systems of Grassmann manifolds over residue-class rings modulo powers \(\mathfrak p\), and we develop a technique for transforming such decompositions into triangular form.

REFERENCES

1. I. Satake, Inst. Hautes Études Sci., Publ. Math., No. 18 (1963).
2. T. Tamagawa, Ann. Math., 77, 387 (1963).
3. G. Shimura, Proc. Nat. Acad. Sci. U.S.A., 49, 824 (1963).
4. A. N. Andrianov, DAN, 177, No. 4, 755 (1967).
5. E. Hecke, Math. Ann., 114, 1 (1937).