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On simple Igusa local zeta functions
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## On simple Igusa local zeta functions

### Roland Martin

**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 $\mu_{c}$ in terms of a suitably normalized Haar
measure $\mu$ and density function $\Phi$. The relevance of these
results in the general theory of Igusa local zeta functions is also
discussed.

*Copyright American Mathematical Society 1996
*

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#### Article Info

- ERA Amer. Math. Soc.
**01** (1995), pp. 108-111
- Publisher Identifier: S 1079-6762(95)03003-6
- 1991
*Mathematics Subject Classification*. Primary 11D79.
*Key words and phrases*. Local zeta functions, canonical bases
- Received by the editors September 23, 1995
- Communicated by David Kazhdan
- Comments (When Available)

**Roland Martin**

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

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

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