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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