## Clifford quartic forms and local functional equations of non-prehomogeneous type

JMS
Vol. 23 (2016), No. 4, Page 791–866.

Kogiso, Takeyoshi ; Sato, Fumihiro
Clifford quartic forms and local functional equations of non-prehomogeneous type
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]

Abstract:
It is known that one can associate local zeta functions satisfying a functional equation to the irreducible relative invariant of an irreducible regular prehomogeneous vector space. We construct polynomials of degree $4$ (called $\it{Clifford}$ $\it{quartic}$ $\it{forms}$) that cannot be obtained from prehomogeneous vector spaces, but for which one can associate local zeta functions satisfying functional equations. The Clifford quartic form is defined for each finite dimensional representation of the tensor product of the Clifford algebras of two positive definite real quadratic forms and cannot be a relative invariant of any prehomogeneous vector space except for a few low dimensional cases. We also classify the exceptional cases of small dimension, namely, we determine all the prehomogeneous vector spaces with Clifford quartic forms as a relative invariant.

Keywords: Clifford quartic forms, local functional equations, prehomogeneous vector spaces

Mathematics Subject Classification (2010): Primary 11S40, 11E45, 11E88; Secondary 15A63, 15A66.
Mathematical Reviews Number: MR3588263

Received: 2015-10-05