Conjugacy classes of centralizers in unitary groups

2.1 Self-U-reciprocal polynomials

Let f⁢(x)=∑i=0dai⁢xi be a polynomial in 𝔽⁢[x] of degree d. We extend the involution on 𝔽 to that of 𝔽⁢[x] by

Let f⁢(x) be a polynomial with f⁢(0)≠0. The corresponding U-reciprocal polynomial of f⁢(x) is a degree d polynomial defined by

A monic polynomial f⁢(x) with a non-zero constant term is said to be self-U-reciprocal if f⁢(x)=f~⁢(x). In terms of roots, it means that for a self-U-reciprocal polynomial, whenever λ is a root, λ¯-1 is also a root with the same multiplicity. Note that

and if f⁢(x)=f1⁢(x)⁢f2⁢(x), then f~⁢(x)=f1~⁢(x)⁢f2~⁢(x). Also, f⁢(x) is irreducible if and only if f~⁢(x) is irreducible. In the case of f⁢(x)=(x-λ)n, the polynomial f⁢(x) is self-U-reciprocal if and only if λ⁢λ¯=1. Over a finite field, we have the following result due to Ennola (see [8, Lemma 2]):

Let T∈GL⁢(V) and let f⁢(x) be its minimal polynomial; then f~⁢(x) is the minimal polynomial of T¯-1. If T∈U⁢(V,B), then its minimal polynomial is monic with a non-zero constant term and is self-U-reciprocal. Let f⁢(x) be a self-U-reciprocal polynomial. We can write it as follows:

where pi⁢(x) is irreducible, self-U-reciprocal, and qj⁢(x) is irreducible, not self-U-reciprocal for all i and j.

2.2 Space decomposition with respect to a unitary transformation

Let T∈U⁢(V,B) and let f⁢(x) be its minimal polynomial. Write f⁢(x) as in equation (2.1). This gives a primary decomposition of V. Furthermore, we have the following:

The proof of this is similar to the orthogonal case as in [11, Section 3] and hence we skip the details. This decomposition helps us reduce the questions about conjugacy classes and z-classes of a unitary transformation to those unitary transformations with minimal polynomial of one of the following two kinds:

1. Type 1: p⁢(x)m, where p⁢(x) is monic, irreducible, self-U-reciprocal polynomial with a non-zero constant term,

2. Type 2: (q⁢(x)⁢q~⁢(x))m, where q⁢(x) is monic, irreducible, not self-U-reciprocal polynomial with a non-zero constant term.

Proposition 4 above gives us a primary decomposition of V into T-invariant B non-degenerate subspaces

where Vi corresponds to the polynomials of Type 1 and Vj=Vqj+Vqj~ corresponds to the polynomials of Type 2. Denote the restriction of T to each Vr by Tr. Then the minimal polynomial of Tr is one of the two types. It turns out that the centralizer of T in U⁢(V,B) is

where Br is a hermitian form obtained by restricting B to Vr. The direct product here comes from the primary decomposition (see [22, Chapter IV, Section 2.8]). Thus the conjugacy class and the z-class of T is determined by the restriction of T to each of the primary subspaces. Hence it is enough to determine the conjugacy class and the z-class of T∈U⁢(V,B), which has minimal polynomial of one of the types listed above.

2.3 Conjugacy classes and centralizers

Let us define

an 𝔽 algebra. Then V is an ET-module, where x acts via T. To keep track of the action we denote this module by VT although it is simply V as an 𝔽-vector space. The ET-module structure on VT determines the GLn-conjugacy class of T. To determine the conjugacy class of T within U⁢(V,B), Springer and Steinberg (see [22, Chapter IV, Section 2.6]) defined a non-degenerate hermitian form HT on VT induced from B and T. We briefly recall this here. As f⁢(x) is a self-U-reciprocal polynomial, there exists a unique involution α on ET such that α⁢(x)=x-1 and α is an extension of σ on scalars. Thus, (ET,α) is an algebra with involution. Further, there exists an 𝔽-linear function lT:ET→𝔽 such that the symmetric bilinear form lT¯:ET×ET→𝔽 given by lT¯⁢(a,b)=lT⁢(a⁢b) is non-degenerate with lT⁢(α⁢(a))=lT⁢(a) for all a∈ET. The hermitian form HT on ET-module VT (with respect to α) satisfies B⁢(e⁢u,v)=lT⁢(e⁢HT⁢(u,v)) for all e∈ET and u,v∈VT and is non-degenerate. Then (see [22, Chapter IV, Sections 2.7–2.8]):

We can decompose the algebra ET as a direct sum of subalgebras with respect to α as ET=E1⊕E2⊕⋯⊕Er, where each component Ei is α-indecomposable (see [22, Chapter IV, Section 2.2]). That is, Ei cannot be further written as a direct sum of α-stable subalgebras. Now denote the restriction of α to Ei by αi. The αi, thus obtained, is again an involution on Ei. Clearly, each Ei is of one of the following types (recall the decomposition of f⁢(x) in equation (2.1), also see [1, Lemma 2.6]):

1. 𝔽⁢[x]〈p⁢(x)d〉, where p⁢(x) is a monic, irreducible, self-U-reciprocal polynomial.

2. 𝔽⁢[x]〈q⁢(x)d〉⊕𝔽⁢[x]〈q~⁢(x)d〉, where q⁢(x) is a monic, irreducible but not self-U-reciprocal.

In the second case, the two components 𝔽⁢[x]/〈q⁢(x)d〉 and 𝔽⁢[x]/〈q~⁢(x)d〉 are isomorphic local rings (induced by the U-reciprocal polynomial structure) and the restriction of α is given by α⁢(a,b)=(b,a) via the fixed isomorphism. Using Wall’s approximation theorem (see Corollary 7 in the next section) it is easy to see that all non-degenerate hermitian forms over such rings are equivalent. Thus to determine equivalence of HT we need to look at modules over rings of Type 1.

2.4 Wall’s approximation theorem

We have reduced the conjugacy problem to equivalence of hermitian forms over certain rings. However, it turns out (see the corollary below) that the second case is easy. Let R be a commutative ring, where 2 is invertible (so that it satisfies the trace condition), let J be its Jacobson radical, and let α be an involution on R. Let M be a finitely generated module over R and (M,B) a non-degenerate hermitian space. We define M¯:=MJ⁢M a module over R¯:=RJ. Now B induces a hermitian form B¯ on M¯ with respect to the involution α¯ of R¯ induced by α. Then it is easy to see that (M¯,B¯) is non-degenerate. The converse is a theorem of Wall (see [27, Theorem 2.2.1], and also [1, Proposition 2.5]) which would be useful for further analysis. For the general theory of hermitian forms over a ring (which need not be commutative) we refer the reader to [20, Chapter 7, Theorem 4.4].

We need the following:

2.5 Unipotent elements

We look at Type 1 more closely, where the minimal polynomial is p⁢(x)d with p⁢(x) an irreducible, self-U-reciprocal polynomial. This includes unipotent elements. The theory of rational canonical forms, which determines conjugacy classes in the group GL⁢(V), gives a decomposition of V=⊕i=1kVdi with 1≤d1≤d2≤…≤dk=d and each Vdi is a free module over the 𝔽-algebra 𝔽⁢[x]/〈p⁢(x)di〉 (see [22, Chapter IV, Section 2.14]). Thus:

This gives us the following:

3.1 Semisimple z-classes

Let T∈U⁢(V,B) be a semisimple element. First, we begin with a basic case.

Now for the general case, let T∈U⁢(V,B) be a semisimple element with minimal polynomial written as in equation (2.1)

where the pi⁢(x) are irreducible, self-U-reciprocal polynomials of degree mi and the qj⁢(x) are irreducible but not self-U-reciprocal of degree lj. Let the characteristic polynomial of T be

Let us write the primary decomposition of V with respect to mT into T-invariant subspaces as

Denote by Ei=𝔽⁢[x]/〈pi⁢(x)〉 and Kj=𝔽⁢[x]/〈qj⁢(x)〉 the field extensions of 𝔽 of degree mi and lj, respectively.

3.2 Proof of Theorem 2

It is already known that the number of z-classes in GLn⁢(𝔽) is finite when 𝔽 has the property FE (see [16, p. 323, (iv)]). The number of conjugacy classes of centralizers of semisimple elements is finite; this follows from Corollary 3. Hence, up to conjugacy, there are finitely many possibilities for 𝒵U⁢(V,B)⁢(s) for s semisimple in U⁢(V,B). Let T∈U⁢(V,B); then it has a Jordan decomposition

Recall that 𝒵U⁢(V,B)⁢(T)=𝒵U⁢(V,B)⁢(Ts)∩𝒵U⁢(V,B)⁢(Tu) and Tu∈ZU⁢(V,B)⁢(Ts)∘ (see [22, Chapter V, Section 3.16]). Now, 𝒵U⁢(V,B)⁢(Ts) is a product of certain unitary groups and general linear groups possibly over a finite extension of 𝔽 (e.g. Ei and Kj in Theorem 2). Since 𝔽0 has the property FE, any finite extension of it has this property. Thus we can apply Corollary 9 to the group 𝒵U⁢(V,B)⁢(Ts) and get that, up to conjugacy, Tu has finitely many possibilities in 𝒵U⁢(V,B)⁢(Ts). Hence, up to conjugacy, 𝒵U⁢(V,B)⁢(T) has finitely many possibilities in U⁢(V,B). Therefore the number of z-classes in U⁢(V,B) is finite.

4.1 z-classes in general linear group

Let p⁢(n) denote the number of partitions of n with generating function

Let z𝔽⁢(n) denote the number of z-classes in GLn⁢(𝔽) and let the generating function be

Let us denote a partition of n by (1k1⁢2k2⁢…⁢nkn) with n=∑ii⁢ki. For convenience, we denote this by n⊢(1k1⁢2k2⁢…⁢nkn). The following result is a consequence of the theory of Jordan canonical forms and the formula is given in [16, Section 10, The absolute case]. However, the formula there has a printing error.

Green computed the number of z-classes in GLn⁢(q) (see [12, Section 1]), which is the function t⁢(n) there. We list them here in our notation (see [16, Section 10, General case]).

Thus if we treat z𝔽q⁢(x)=∏i=1∞z⁢(xi), this proposition beautifully reflects the arithmetic nature of the field. To compare these numbers we make a table for small rank. The last row of this table is there in the work of Green.

4.2 z-classes in hyperbolic unitary group

In geometry the unitary groups used are over ℂ. Let V be a vector space over ℂ of dimension n+1. Hermitian forms are classified by signature (as in the case of quadratic forms over ℝ) and the corresponding groups are denoted as U⁢(r,s), where r+s=n+1. The group given by the identity matrix is the compact unitary group denoted as Un+1⁢(In+1)=U⁢(n+1,0). The genus number (which is the number of z-classes) of the compact special unitary group has been computed in [3, Theorem 3.1]. We record the result here as follows:

The centralizers and z-classes in the group U⁢(n,1) have been described by Cao and Gongopadhyay in [5, Section 4]. However, they have not done explicit counting. Keeping in mind the spirit of this section, we enumerate the number of z-classes in this group as well. We briefly recall the notation here and urge the interested reader to see the source and references therein for details. Let V be an (n+1)-dimensional vector space over ℂ with the hermitian form given by

which is of signature (n,1). The unitary group is

Let V- be the vectors of negative length in V. The image of V- in the projective space ℙ⁢(V) is denoted as ℌℂn which is a complex hyperbolic space. An element of the group is called elliptic if it has a fixed point on ℌℂn and is said to be parabolic, respectively, hyperbolic, if it has exactly one, respectively, two fixed points on the boundary of ℌℂn. Every element falls in one of these three classes. Using the conjugation classification, we know that if an element g∈U⁢(n,1) is elliptic or hyperbolic, then it is semisimple. But a parabolic element need not be semisimple. However, it has a Jordan decomposition g=gs⁢gu, where gs is elliptic, hence semisimple, and gu is unipotent. In particular, if a parabolic isometry is unipotent, then it has minimal polynomial (x-1)2 or (x-1)3 and is called a vertical translation or non-vertical translation respectively. The centralizers of these elements are described in [5, Corollary 1.2] and we present the counting for z-classes here.