ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 1, Issue 3, 1995

ON SIMPLE IGUSA LOCAL ZETA FUNCTIONS

ROLAND MARTIN

(Communicated by David Kazhdan)

Abstract. The objective of this announcement is the statement of some recent results on the classification of generalized Igusa local zeta functions associated to irreducible matrix groups. The definition of a simple Igusa local zeta function will motivate a complete classification of certain generalized Igusa local zeta functions associated to simply connected simple Chevalley groups. In addition to the novelty of these results are the various methods used in their proof. These methods include use of the concept of canonical basis from quantum group theory and a formula expressing Serre’s canonical measure μ_{c} in terms of a suitably normalized Haar measure μ and density function Φ. The relevance of these results in the general theory of Igusa local zeta functions is also discussed.

1. Introduction

The study of local zeta functions of Igusa type commenced in earnest in 1974 in [3]. Through the work of various people was developed the concept of a generalized Igusa local zeta function. For ease of explication, only the specialization of this concept of interest here is introduced. The reader is referred to [5] or [4] for a more general definition.

Let K be a finite algebraic extension of Q_{p}. Let O_{K}, πO_{K} denote the ring of integers of K, ideal of nonunits of O_{K}, respectively. Set card(O_{K}/πO_{K}) = q. For technical reasons assume 2 /| q (see [5]). Let | |_{K} be the absolute value on K normalized as |π|_{K} = q^{−1}. Let G′ be a simply connected simple Chevalley Kgroup and ρ a finite dimensional K-representation of G′. The almost direct product G = ρ(G′)(G_{m}1_{dim ρ}), where G_{m} denotes the algebraic group defined by K^{∗} and 1_{dim ρ} the dim ρ×dim ρ identity matrix, is a K-subgroup of GL_{dim ρ}, not contained in SL_{dim ρ}. Set T as the maximal K-split torus in G, f as the positive generator of Hom(G, G_{m}) and G^{0} = G_{K} ⋂ Mat_{dim ρ}(O_{K}) where G_{K} denotes the K-rational points of G.

\begin{matrix}c \\ ◯ \end{matrix}1995 American Mathematical Society

Received by the editors September 23, 1995.

1991 Mathematics Subject Classification. Primary 11D79.

Key words and phrases. Local zeta functions, canonical bases.

Definition 1.1. The generalized Igusa local zeta function associated to (G′, ρ) is

Z_{K}(s) = ∫_{G^{0}} |f(g)|_{K}^{s}μ_{c}(g)

where μ_{c} is Serre’s canonical measure defined on G_{K} and s ∈ C, Re(s) > 0.

Typically, one would like to compute a finite form for Z_{K}(s) and examine this finite form for further relations. For example, the generalized Igusa local zeta function associated to (SL_{1}, 1) has finite form Z(q^{−1}, q^{−s}) = (1 − q^{−1})/(1 − q^{−s−1}) with relation

Z(q^{−1}, q^{−s})|_{q↦q^{−1}} = q^{−s}Z(q^{−1}, q^{−s}) .

Note that this generalized Igusa local zeta function is universal in the variables q^{−1}, q^{−s} (see [4]). There is a criterion for answering the typical questions in the case of a generalized Igusa local zeta function associated to (G′, ρ).

Since the measure μ_{c} is difficult to work with, define a function Φ on G_{K} by

Φ(g) = \frac{μ_{c}(g(G_{K} ⋂ GL_{dim ρ}(O_{K})))}{μ(G_{K} ⋂ GL_{dim ρ}(O_{K}))}

where μ is the Haar measure on G_{K} normalized to be canonical measure on the group G_{K} ⋂ GL_{dim ρ}(O_{K}). It follows that μ_{c}(g) = Φ(g)μ(g) and, thus, that

Z_{K}(s) = ∫_{G^{0}} |f(g)|_{K}^{s}Φ(g)μ(g) .

The details are in [5]. Via the p-adic Bruhat decomposition for G_{K}, the computation of Φ(g) is reduced to that of Φ(ξ(π)) where ξ ∈ Hom(G_{m}, T) is as in [4] or [5].

Definition 1.2. The generalized Igusa local zeta function Z_{K}(s) associated to (G′, ρ) is simple if Φ is independent of ξ.

Remark on Definition 1.2. As just explained above, Z_{K}(s) = Z_{K}(s, Φ). Therefore, the property of Z_{K}(s) being a simple generalized Igusa local zeta function depends only on Z_{K}(s) itself.

Criterion ([4]). If Z_{K}(s) is simple, then Z_{K}(s) has a finite form that expresses Z_{K}(s) as a rational function Z(q^{−1}, q^{−s}) satisfying the functional equation

Z(q^{−1}, q^{−s})|_{q↦q^{−1}} = q^{−deg(f)s}Z(q^{−1}, q^{−s}).  •

Remarks on the Criterion. Note first that Z_{K}(s) is universal in the variables q^{−1}, q^{−s}. Note also that Φ is not “residual” as defined by J. Denef and D. Meuser because Φ(g) does not only depend on g mod πO_{K}. Therefore, it is not immediate that Z_{K}(s) has either a finite form or the functional equation • (see [1]).

2. Results

There is now motivation for classifying as simple or not the generalized Igusa local zeta functions associated to (G′, ρ). By assumption, G′ is in one of the families of nine simply connected simple Chevalley K-groups distinguished by their respective Lie algebras A_{ℓ}, B_{ℓ}, C_{ℓ}, D_{ℓ}, E_{6}, E_{7}, E_{8}, F_{4} and G_{2}. Denote by LG′ the Lie algebra of G′. It is well known that the nodes of the Dynkin diagram of LG′ may be labeled canonically by the fundamental representations of G′. We denote these fundamental representations by π_{i}, 1 ≤ i ≤ rk LG′. Recall that ρ has a unique expression in terms of the fundamental representations π_{i} as

ρ = ^{{rk LG′ \atop ∗} ; i=1} π_{i}^{n_{i}}

where ∗ denotes the Cartan product. We are able to parametrize the simple generalized Igusa local zeta functions associated to (G′, ρ) by examining the above Cartan product decomposition of ρ.

Therefore, orient the Dynkin diagrams of the simple Lie algebras as in [8] and label their nodes consecutively with the π_{i} from left to right reserving for a branched node the last index rk LG′. For example, the Dynkin diagram for D_{ℓ} would be labeled

[figure: π_{1}; π_{2}; π_{3}; _{. . . . . .}; π_{ℓ−3}; π_{ℓ−2}; π_{ℓ}; π_{ℓ−1}]

Definition 2.1. Given (G′, ρ), the subgraph of the Dynkin diagram of G′ determined by the elimination of any nodes (and their incident edges) associated to fundamental representations π_{i} of G′ not occurring in the Cartan product decomposition of ρ is called the decomposition diagram.

Our result is the following.

Theorem 2.1 ([7]). Let G′ be of type A_{ℓ} (ℓ ≥ 1), B_{ℓ} (ℓ ≥ 2), C_{ℓ} (ℓ ≥ 3), D_{ℓ} (ℓ ≥ 4), E_{6}, E_{7}, E_{8}, F_{4} or G_{2}. The generalized Igusa local zeta function Z_{K}(s) associated to (G′, ρ) is simple if and only if the decomposition diagram of (G′, ρ) is connected.

Thus, in the case of G′ of type D_{ℓ} and ρ = i=1^{{ℓ \atop ∗}}π_{i}^{n_{i}} , the corresponding Z_{K}(s) is simple unless one or more of the following conditions hold:

(1) there exist n_{i′}, n_{j′} nonzero, 1 ≤ i′ < j′ ≤ ℓ − 2, such that j′ − i′ > 1 and n_{i′+1} = … = n_{j′−1} = 0, or

(2) there exists n_{i′} nonzero, 1 ≤ i′ < ℓ − 2, such that n_{ℓ−1}, n_{ℓ} or n_{ℓ−1} and n_{ℓ} are nonzero and n_{i′+1} = … = n_{ℓ−2} = 0.

The main intermediary result, Theorem 2.2 below, is proved using the concept of canonical basis from quantum group theory. The reader is referred to [2] for background.

Theorem 2.2 ([7]). Let ρ be a finite dimensional irreducible K-representation of G′. Suppose the Z_{K}(s) corresponding to (G′, ρ) is simple. Then the Z_{K}(s) corresponding to (G′, n_{∗} ; i=1ρ) is simple.

Remark on the proof of Theorem 2.2. The proof is an induction argument taking advantage of various properties of the canonical basis of the representation space V of ρ.

Use of Theorem 2.2 allows one to prove Theorem 2.1 via case-by-case examination.

One can see that most choices of (G′, ρ) yield non-simple Z_{K}(s). There are some isolated results related to determining the finite form and verifying the functional equation • for Z_{K}(s) in the non-simple case (see [6]). It is, however, unknown which of these Z_{K}(s) have finite forms and satisfy • .

References

[1] J. Denef and D. Meuser, A functional equation of Igusa’s local zeta function, AJM 113 (1991), pp. 1135-1152.

[2] M. Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), pp. 249-260.

[3] J. Igusa, Complex powers and asymtotic expansions I, J. reine. angew. Math. 268/269 (1974), pp. 110-130; ibid 278/279 (1975), pp. 307-321.

[4] J. Igusa, Universal p-adic zeta functions and their functional equations, AJM 111 (1989), pp. 671-716.

[5] R. Martin, On generalized Igusa local zeta functions associated to simple Chevalley K-groups of type A_{ℓ}, B_{ℓ}, C_{ℓ}, D_{ℓ}, E_{6}, E_{7}, E_{8}, F_{4} and G_{2} under the adjoint representation, preprint 1992.

[6] ———, The universal p-adic zeta function associated to the adjoint group of SL_{ℓ+1} enlarged by the group of scalar multiples, preprint 1992.

[7] ———, On the classification of Igusa local zeta functions associated to certain irreducible matrix groups, preprint 1995.

[8] V. Varadarajan, Lie groups, Lie algebras, and their representations, Springer-Verlag, New York, 1984.

Department of Mathematics, United States Naval Academy, Annapolis, MD 21114

E-mail address: rem@sma.usna.navy.mil