Root Sublattices and Torus-Restriction of Prehomogeneous Vector Spaces Associated with Dynkin Quivers

Abstract

For a Dynkin quiver Γ with r vertices, a subset S of the vertices of Γ, and an r-tuple d = (d⁽¹⁾, d⁽²⁾,…, d^(r)) of positive integers, we define a “torus-restricted” representation (G_S, R _d (Γ)) in natural way. Here we put G_S = G₁ × G₂ × … ×G_r, where each G_i is either SL(d⁽ⁱ⁾) or GL(d⁽ⁱ⁾) according to S containing i or not. In this paper, for a prescribed torus-restriction S, we give a necessary and sufficient condition on d that R _d (Γ) has only finitely many G_S-orbits. This can be paraphrased as a condition whether or not d is contained in a certain lattice spanned by positive roots of Γ. We also discuss the prehomogeneity of (G_S, R _d (Γ)).

Key Words:

• Dynkin quiver
• Root sublattice
• Semisimple finite prehomogeneous vector space
• Tilting module

2010 Mathematics Subject Classification:

• Primary 14L30
• Secondary 16G20, 11S90

ACKNOWLEDGMENT

The authors must give an address of thanks to Dr. Hiroshi Nagase at Tokyo Gakugei University, who instructed the second author in useful results on tilting modules mentioned in Section 4.

Notes

Communicated by K. Misra.