Prehomogeneous vector space

In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space "V" together with a subgroup "G" of GL("V") such that "G" has an open dense orbit in "V". Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and analysis, as well as representation theory. The irreducible PVS were classfied by Sato and T. Kimura in 1977, up to a transformation known as "castling". They are subdivided into two types, according to whether the semisimple part of "G" acts prehomogeneously or not. If it doesn't then there is a homogeneous polynomial on "V" which is invariant under the semisimple part of "G".

etting

In the setting of Sato, "G" is an algebraic group and "V" is a rational representation of "G" which has a (nonempty) open orbit in the Zariski topology. However, PVS can also be studied from the point of view of Lie theory: for instance, in Knapp (2002), "G" is a complex Lie group and "V" is a holomorphic representation of "G" with an open dense orbit. The two approaches are essentially the same, and it is also interesting to study the theory over the real numbers. We assume, for simplicity of notation, that the action of "G" on "V" is a faithful representation. We can then identify "G" with its image in GL("V"), although in practice it is sometimes convenient let "G" be a covering group.

Although prehomogeneous vector spaces do not necessarily decompose into direct sums of irreducibles, it is natural to study the irreducible PVS (i.e., when "V" is an irreducible representation of "G"). In this case, a theorem of Élie Cartan shows that

:"G" ≤ GL("V")

is a reductive group, with a centre that is at most one-dimensional. This, together with the obvious dimensional restriction

:dim "G" ≥ dim "V",

is the key ingredient in the Sato–Kimura classification.

Castling

The classification of PVS is complicated by the following fact. Suppose "m" > "n" > 0 and "V" is an "m"-dimensional representation of "G" over a field IF. Then::$(G\; imes\; SL(n),\; Votimesmathbb\; F^n)$ is a PVS if and only if $(G\; imes\; SL(m-n),\; V^*otimes\; mathbb\; F^\{m-n\})$ is a PVS.The proof is to observe that both conditions are equivalent to there being an open dense orbit of the action of "G" on the Grassmannian of"n"-planes in "V", because this is isomorphic to the Grassmannian of ("m"-"n")-planes in "V"^*.

(In the case that "G" is reductive, the pair ("G","V") is equivalent to the pair ("G", "V"^*) by an automorphism of "G".)

This transformation of PVS is called castling. Given a PVS "V", a new PVS can be obtained by tensoring "V" with IF and castling. By repeating this process, and regrouping tensor products, many new examples can be obtained, which are said to be "castling-equivalent". Thus PVS can be grouped into castling equivalence classes. Sato and Kimura show that in each such class, there is essentially one PVS of minimal dimension, which they call "reduced", and they classify the reduced irreducible PVS.

Classification

The classification of irreducible reduced PVS ("G","V") splits into two cases: those for which "G" is semisimple, and those for which it is reductive with one-dimensional centre. If "G" is semisimple, it is (perhaps a covering of) a subgroup of SL("V"), and hence "G"×GL(1) acts prehomogenously on "V", with one-dimensional centre. We exclude such trivial extensions of semisimple PVS from the PVS with one-dimensional center. In other words, in the case that "G" has one-dimensional center, we assume that the semisimple part does "not" act prehomogeneously; it follows that there is a "relative invariant", i.e., a function invariant under the semisimple part of "G", which is homogeneous of a certain degree "d".

This makes it possible to restrict attention to semisimple "G" ≤ SL("V") and split the classification as follows:
# ("G","V") is a PVS;
# ("G","V") is not a PVS, but ("G"×GL(1),"V") is.

However, it turns out that the classification is much shorter, if one allows not just products with GL(1), but also with SL("n") and GL("n"). This is quite natural in terms of the castling transformation discussed previously. Thus we wish to classify irreducible reduced PVS in terms of semisimple "G" ≤ SL("V") and "n" ≥ 1 such that either:
# $(G\; imes\; SL(n),Votimes\; mathbb\; F^n)$ is a PVS;
# $(G\; imes\; SL(n),Votimes\; mathbb\; F^n)$ is not a PVS, but $(G\; imes\; GL(n),Votimes\; mathbb\; F^n)$ is.

In the latter case, there is a homogeneous polynomial which separates the "G"×GL("n") orbits into "G"×SL(n) orbits.

This has an interpretation in terms of the grassmannian Gr_"n"("V") of "n"-planes in "V" (at least for "n" ≤ dim "V"). In both cases "G" acts on Gr_"n"("V") with a dense open orbit "U". In the first case the complement Gr_"n"("V")-"U" has codimension ≥ 2; in the second case it is a divisor of some degree "d", and the relative invariant is a homogeneous polynomial of degree "nd".

In the following, the classification list will be presented over the complex numbers.

General examples

Other Hermitian symmetric spaces yields prehomogeneous vector spaces whose generic points define Jordan algebras in a similar way.The Jordan algebra "J"("m"−1) in the last row is the spin factor (which is the vector space R^"m"−1 ⊕ R, with a Jordan algebra structure defined using the inner product on R^"m"−1). It reduces to $J\_2(mathbb\; R),\; J\_2(mathbb\; C),\; J\_2(mathbb\; H),J\_2(mathbb\; O)$ for "m"= 3, 4, 6 and 10 respectively.

The relation between hermitian symmetric spaces and Jordan algebras can be explained using Jordan triple systems.

References

* Anthony Knapp, "Lie Groups Beyond an Introduction" [http://www.math.sunysb.edu/~aknapp/books/beyond2.html] , 2nd Edition, Progress in Mathematics, volume 140, Birkhäuser, Boston, 2002. See Chapter X.
* Mikio Sato and T. Kimura, "A classification of irreducible prehomogeneous vector spaces and their relative invariants" [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.nmj/1118796150] , Nagoya Mathematical Journal, volume 65 (1977), 1-155.
* R. W. Richardson, Jr, "Conjugacy Classes in Parabolic Subgroups of Semisimple Algebraic Groups" [http://dx.doi.org/10.1112/blms/6.1.21] , Bull. London Math. Soc., volume 6 (1974) 21-24.
* E. B. Vinberg, "On the classification of the nilpotent elements of graded Lie algebras", Soviet Math Doklady, volume 16 (1975) 1517-1520.