# Introduction to Prehomogeneous Vector Spaces by Tatsuo Kimura

By Tatsuo Kimura

This can be the 1st introductory booklet at the concept of prehomogeneous vector areas, brought within the Nineteen Seventies through Mikio Sato. the writer was once an early and critical developer of the speculation and remains to be lively within the box. the topic combines components of numerous components of arithmetic, resembling algebraic geometry, Lie teams, research, quantity idea, and invariant conception. a major aim is to create functions to quantity idea. for instance, one of many key subject matters is that of zeta capabilities hooked up to prehomogeneous vector areas; those are generalizations of the Riemann zeta functionality, a cornerstone of analytic quantity conception. Prehomogeneous vector areas also are of use in illustration idea, algebraic geometry and invariant idea. This publication explains the fundamental ideas of prehomogeneous vector areas, the elemental theorem, the zeta services linked to prehomogeneous vector areas, and a type idea of irreducible prehomogeneous vector areas. It strives, and to a wide volume succeeds, in making this content material, that's by way of its nature quite technical, self-contained and obtainable. the 1st element of the e-book, ""Overview of the idea and contents of this book,"" is very noteworthy as a great creation to the topic.

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We call the isotropy subgroup Gv = {g G G; p(g)v = u} at a generic point v a generic isotropy subgroup. We see that generic isotropy subgroups are isomorphic to each other, since Gp(p)v = gGvg~l (g G G). , p(9~ 1 h'd)v = v • So we have g~l hg G Gv, which is equivalent to h G gGvg ~l . 3. If a triplet (G ,p,V ) is a prehomogeneous vector space, then dim G ^ dim V. Hence if dimG < dim V, then (G,p, V) is not a prehomogeneous vector space. P r o o f . 2, p(G)v = V implies d im G - dim V = d im G v ^ 0, so dim G ^ dim V.

N by T(g) = gx (g e GLn). n at the unit elem ent is given by dT(A) = Ax (A e gin), because d T ^ )* = y OZiitaXi ^ P=/n v AuXj. I Let G C GLm and n = dim V. For x G V, we define a mapping F = T o p : G GL(V) ^ V by F(<7) = p(g)x . Then the differential mapping dF : g = TeG ^ dT 0[(V) —> V at the unit element is given by dF(A) = dp(A)x, and we have ( 2. 1) E h3= 1 d{p{g)x)k dgij ■Aij = ((ip(A)x)k (1 ^ k ^ n). 9=Im Further we define a mapping H : G -* Q by H(g) = f(p(g)x) = x(g)f(x)- The differential mapping dH : g = Te(G) -*• 12 = at the unit element e = Im is given by (dH)(A) = d x(A )f(x).

Then the differential mapping dF : g = TeG ^ dT 0[(V) —> V at the unit element is given by dF(A) = dp(A)x, and we have ( 2. 1) E h3= 1 d{p{g)x)k dgij ■Aij = ((ip(A)x)k (1 ^ k ^ n). 9=Im Further we define a mapping H : G -* Q by H(g) = f(p(g)x) = x(g)f(x)- The differential mapping dH : g = Te(G) -*• 12 = at the unit element e = Im is given by (dH)(A) = d x(A )f(x). E(E d(p(g)g)fc % 5 71 ft £ = X j ¿ÿjH*) ' (dP(A )x )k =