Weil Representation of a Generalized Linear Group over a Ring of Truncated Polynomials over a Finite Field Endowed with a Second Class Involution

We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G={\rm SL}_*^{1}(2,A_n)$ ($A_n=K[x]/\langle x^n\rangle$, $K$ the quadratic extension of the finite field $k$ of $q$ elements, $q$ odd), where $A_n$ is endowed with a second class involution. After the construction of a specific data, the representation is defined on the generators of a Bruhat presentation of $G$, via linear operators satisfying the relations of the presentation. The structure of a unitary group $U$ associated to $G$ is described. Using this group we obtain a first decomposition of $\rho$.


Introduction
Weil representations is a central topic in representation theory. They arise as a consequence of one of the many seminal works of A. Weil [17]. They are projective representations of the groups Sp(2n, F ), F a locally compact field. These representations are an important subject of study. By decomposition into irreducible factors, Weil representations provide all irreducible complex representations of the general linear group over a finite field, and over a local field of residual characteristic different from 2. It has many other important consequences, and it has applications in different topics as theta functions and physics to mention only some of them. In fact, in representation theory it helps to understand (among other things) the harmonic analysis of the group, and in the Langlands program, to explain the relations between linear groups defined over local or global fields, and the Galois groups of the fields.
A point of view that has been favorably used in representation theory, is to extend to higher rank groups, methods successfully used in lower rank groups. This philosophy has led Pantoja and Soto-Andrade (see [11,12,13]) to define the groups GL ε * (2, A) and SL ε * (2, A), where ε = ±1. These are "generalized linear groups" of rank 2 with coefficients in a unitary involutive ring (A, * ) [10] (ε = ±1). In this way, the symplectic group and the orthogonal group of rank n over a field F appear as groups SL ε * (2, A) considering as involutive ring, the ring of n × n matrices over F with the transposition of matrices, and taking, respectively, ε equals to −1 and 1. Different choices of involutive rings produce new examples of diverse kind.
The rings considered are, in general, non-commutative, and this non-commutativity is controlled by the relation (ab) * = b * a * . In this sense, it should be pointed out the similarity of our group GL * (2, A), with the quantum group GL q (2) (as a "group" of matrices with some commuting relations). Furthermore, the generalized linear groups afford the notion of a * -determinant, det * , in analogy with the q-determinant. Also, the homomorphism * -determinant can be considered as a first way of describing the mentioned groups, as these can be defined as two by two matrices with coefficients in A, (satisfying some commutation relations that involve * ), with * -determinant different from 0. Then, the groups SL * (2, A) appear as the kernel of the homomorphism det * .
The approach, used here, has been considered in diverse interesting groups [6,7,16]. Having a presentation of the group, that the authors call Bruhat presentation, and certain specific data, a generalized Weil representation has been constructed. Historically, the groups G were defined first for ε = −1. The work of Soto-Andrade on the symplectic group over a finite field [15] and the representations constructed by Gutiérrez [6] appear as examples of these constructions in [7]. Later, after the generalization to ε = 1, Vera [16] constructed a generalized Weil representation of the orthogonal group over a finite field, where she also relates the representation with the one that can be constructed by the theory of dual pairs of Howe [9]. However, even that the orthogonal group is a natural example of a generalized linear group with ε = 1, her work is done treating the group as a "symplectic type" group, i.e., as a SL −1 * group, by twisting the transposition of matrices to define the involution of the ring.
Our method to construct Weil representations is one of many successful approaches to attack this topic, and has been studied by several authors (see, e.g., [1,4,5,17]). Using Weil's original point of view, Szechtman et al. in [2] construct Weil representations of symplectic groups over finite rings via Heisenberg groups. Recently, Herman and Szechtman [8] construct Weil representations of unitary groups associated to a finite, commutative, local principal ring of odd characteristic, by imbedding the group into a symplectic group.
A great diversity of groups can be originated via generalized linear groups for appropriate choices of involutive rings. For each of them, Weil representations could be constructed by using our approach of defining linear operators for each generator of the group in such a way that they satisfy the basic (universal) relations of a "simple" presentation. This variety of cases for which a (generalized) Weil representation could be produced has led us to verify that, in practice, the procedure is effective. In fact, several examples of groups GL ε * (2, A) and SL ε * (2, A) have been given in loc. cit. for different choices of involutive rings provided with first class involutions [10]. However, no example with an involutive ring (algebra) provided with a second class involution has been considered up to now. Furthermore, explicit constructions for generalized linear groups with ε = 1 have not been made so far.
In this work, we construct a Weil representation of a generalized SL 1 * "orthogonal type group" (i.e., a generalized special linear group with ε = 1) over the ring A n = K[x]/ x n , which is a nonsemisimple ring (algebra in fact) over K, K the quadratic extension of the finite field k of q elements (q odd), provided with a second class involution. We show first that the group under consideration has a Bruhat presentation, after which a data necessary to produce a Weil representation of G via relations and generators, is specifically described. We also obtain a first decomposition of the representation. Using a complete different approach leaning on the works of Amritanshu Prasad and Kunal Dutta [3,14], a further decomposition could be achieved. Furthermore, a comparison of their methods and ours will be performed in a work that will appear elsewhere.
The paper is organized as follows: In Section 2, we present the main definitions on generalized classical groups GL ε * (2, A) and SL ε * (2, A) for an involutive ring (A, * ) and we describe some properties of the truncated polynomials ring K[x]/ x n , for K a quadratic extension of a field of q elements (q odd). In Section 3, a Bruhat presentation of SL 1 * (2, A n ) is constructed. Section 4 is devoted first to recall a very general procedure to construct generalized Weil representations of SL ε * (2, A), for a group with a Bruhat presentation and a suitable data (M, χ, γ, c). After this, the necessary data for the group under consideration is produced and completely described and detailed. Finally, in Section 5, we define the abelian "unitary" group U (M, χ, γ, c) of (M, χ, γ, c), with the explicit decomposition into cyclic subgroups, which allow us to get a first decomposition of the constructed representation.

Preliminaries
In this section, we fix notations and recall some basic facts about generalized general special linear groups over involutive rings.
Let A be a unitary ring endowed with an involution * , i.e., an antiautomorphism a → a * of A of order two. We denote by Z(A) the center of A, and we write A × (respectively Z(A) × ) for the group of invertible elements of A (respectively, of Z(A)). T sym (respectively T asym ) stands for the set of symmetric (respectively antisymmetric) elements of the subset T of A, i.e., the set of elements a in T such that a * = a (respectively a * = −a). The involution * induces an involution on the ring of 2 × 2 matrices with coefficients in A, by (g * ) ij = g * ji (g ∈ M(2, A)), which we denote also with the symbol * .

The groups SL
We give a brief description of the groups SL 1 * (2, A). For more details see [13]. If A is a unitary ring with an involution * , and J = 0 given by det * (g) = λ(g) = ad * + bc * = a * d + c * b is an homomorphism.
We define SL 1 * (2, A) as the kernel of det * . One can observe that the entries of g =

The involutive ring of truncated polynomials
Let k be a finite field of q elements, where q is a power of an odd prime p. We consider K the unique quadratic extension of k and we take ∆ ∈ K such that K = k(∆) and ∆ 2 ∈ k. We write a + b∆ to denote the image a − b∆ of the element a + b∆ under the nontrivial element of the Galois group of the extension K/k. Let which will be considered as polynomials (with coefficients in K), truncated at n (i.e., such that x m = 0 for m n).
We define an involution * in the k-algebra A n by a + b∆ → a + b∆, We first present some results concerning cardinalities about sets that we will use later on.

2.
A sym a i x i : a 2i ∈ k and a 2i+1 ∈ ∆k has cardinality |A sym n | = q n .

3.
A asym Proof . 1. The result is clear observing that an invertible element of the ring must have nonzero constant term.
a i x i : a i ∈ k for i even and a i ∈ ∆k for i odd , and the result follows.
3. Similar to 2. This completes the proof.
Proof . The group SL * (2, A n ) acts on M 2×1 (A n ) by left multiplication. Set We claim that the orbit Orb SL 1 * (2,An) In fact, we note that the first column a c of a matrix in SL 1 Since a or c must be invertible, we get ca −1 (or ac −1 ) is anti-symmetric, then c = ua (or a = uc), for some anti-symmetric This proves the claim. Now given that the sets are disjoint we verify Finally, the isotropy group Stab 1 0 is the group of upper unipotent matrices in SL 1 * (2, A n ) which has cardinality q n , and therefore our proposition follows.

Bruhat presentation for SL
In this section, we prove that the group G = SL 1 * (2, A n ) has a Bruhat-like presentation, which will be used in the construction of a Weil representation of G.
To this end, we set noticing that the matrices h t , ω, u s ∈ SL 1 * (2, A). Lemma 1. Let a and c be two elements in A n such that Aa + Ac = A and a * c = c * a. Then, there exits an element s ∈ A asym n such that a + sc is invertible.
Proof . We first observe that a or c has to be invertible. Suppose that a is invertible, then considering s = 0 we have the result. On the other hand, if a is non-invertible, then c is invertible and so any nonzero element s in k∆ proves the lemma.
From here, SL 1 * (2, A n ) is generated by the matrices h t , u s and w with t ∈ A × n , s ∈ A asym n .
Lemma 2. Let a and b be two non-invertible antisymmetric elements in A n . Then, we can find an antisymmetric invertible element v ∈ A n such that a − v −1 and b + v are antisymmetric invertible elements in A n .
Proof . Since a, b are non-invertible elements, they are in the ideal generated by x. Then taking any nonzero element v ∈ ∆k, we check that a − v −1 and b + v are antisymmetric invertible elements in A n .
Lemmas 1, 2 and Proposition 3 prove, using the same argument as in Theorem 15 of [11], our next result: , with the commutating relations: give a presentation of SL 1 * (2, A n ). f) γ(t, m + z) = γ(t, m)γ(t, z)χ(m, zt) for all m, z ∈ M , t ∈ A asym , where t is anti-symmetric invertible in A and c ∈ C × satisfies c 2 |M | = 1; Then, with the above data we have (see [7]): A) has a Bruhat presentation, the data (M, χ, γ, c) defines a (linear) representation (C M , ρ) of SL 1 * (2, A), which we call Weil representation, by 2) ρ ht (e a ) = α(t)e at −1 , for a ∈ M , b ∈ A asym , t ∈ A × and e a the Dirac function at a, defined by e a (u) = 1 if u = a and e a (u) = 0 otherwise.

Construction of data for SL
Since we already know that SL 1 * (2, A n ) has a Bruhat presentation, we construct now a data for this group, in order to apply Theorem 2.
Let ψ 0 be a nontrivial additive character of K such that ψ 0 is nontrivial in k and in ∆k. We consider the biadittive function χ from A n × A n to C × given by where ψ is the nontrivial character of A n defined as ψ a o + a 1 x + · · · + a n−1 x n−1 = ψ 0 (a n−1 ).
We take M = A n and we assume α = 1. It is clear that a), b) and c) above are fulfilled. We prove now d).
Proof . Let a be a nonzero element of A n . We need to prove that there is an element b in A n such that χ(a, b) = 1. Let us write a = a o + a 1 x + · · · + a n−1 x n−1 . If a i is the first nonzero coefficient of a, set b = tx n−i−1 . Then χ(a, b) = ψ 0 ((−1) iā i t). If t runs over K, then so does (−1) iā i t. Since the character ψ 0 is nontrivial, the result follows.
Proof . By defintion γ(t, y) = χ(− 1 2 ty, y). We set y = and t ∈ A asym × . We will write sometimes (y) i for the coefficient of x i in y. Then Since t * = −t we have λ i = d i ∈ k for i odd, and λ i = d i ∆ ∈ k∆ for i even.
We sum next over α 1 and α n−2 , and we continue with this process to obtain at the end that (1) is (q 2 ) n 2 = q n .

A f irst decomposition
We will get a first decomposition of ρ, constructed in Theorem 2. To this end, we lean on Theorem 7.6 in [7]. The unitary group U = U (χ, γ, c) consisting of all A n -linear automorphism ϕ of M such that γ(b, ϕ(x)) = γ(b, x), for any b ∈ A n and x ∈ M , allows us to obtain a decomposition of the Weil representation. In fact, the characters of U define the invariant subspaces of a decomposition of ρ. So, we will devote ourselves to obtain the structure of this group. Observing first that U is abelian, Theorem 7.6 in [7] reads as: If Λ ∈ U , let W Λ be the vector subspace of W of the Λ-homogeneous functions, that is, the vector subspace of the functions f ∈ W such that f (ua) = Λ(u)f (a), for a ∈ A n and u ∈ U , then Theorem 3. The Weil representation (W, ρ) is the direct sum of all W Λ , where Λ runs over all linear characters of U .
We first prove: Proposition 6. We have 1. The group A × n acts transitively on A sym n ∩ A × n by a · t = ata * . 2. The group of units A × n acts transitively on A asym n ∩ A × n under the same action.
Now, for an arbitrary invertible symmetric element t = a 0 + a 1 ∆x + a 2 x 2 + · · · + a n−1 ∆x n−1 , given that any nonzero element of k is in Orb(1), we have that t = a 0 1 + a −1 0 a 1 ∆x + a −1 0 a 2 x 2 + · · · + a −1 0 a n−1 ∆x n−1 belongs to Orb(1) and therefore Orb(1) = A sym n ∩ A × n . The case when n is odd is handled in a similar way. 2. Notice that ∆a ∈ A asym n ∩ A × n , for any symmetric invertible element a ∈ A n . Then, it follows from part 1 that ∆(A sym n ∩ A × n ) = ∆Orb(1) is a subset of A asym n ∩ A × n that has the same cardinality than A asym n ∩ A × n . We have that the orbit of ∆ is the unique orbit for the action. This proves the proposition.
Next we start to prove that the group U = U (χ, γ, c) is isomorphic to the group of all a ∈ A × n such that aa * = 1. We first prove Lemma 4. The subgroup {a ∈ A n : aa * = 1} has (q + 1)q n−1 elements.
Proposition 8. We have • If n is even, then the group U 0 consists of all elements z of the form: with λ i ∈ k. • If n is odd, then the group U 0 consists of all elements z of the form: Proof . Part 1 follows directly from the definitions.
In general, when i is even, b i +b i is a k-valued function f i (λ 1 , . . . , λ i−1 ), and Im(b i ) = f i−1 (λ 1 , . . . , λ i−1 ) is a new variable, getting b i = λ i . When i is odd, b i −b i determines a k-valued function, we set this time Im(b i ) = f i−1 (λ 1 , . . . , λ i−1 ) and Real(b i ) = f i−1 (λ 1 , . . . , λ i−1 for a new variable independent from λ 1 , . . . , λ i−1 . We can write b i = λ i + f i−1 (λ 1 , . . . , λ i−1 )∆. The result now follows. The field k = F q (where q = p t , p an odd prime number) is a t-dimensional vector space over F p . We set e 1 , . . . , e t for a basis of k over F p .
We describe the elements of U 0 as in Proposition 8, and we define: Definition 1. Let i be relatively prime to p, and l = 1, . . . , t. H i,l denotes the cyclic subgroup of U 0 of order d = d i,l (d is a power of p), generated by z = 1 + e l x i + α 2 x 2i + · · · + α ord(z) x ord(z)i for i odd, and generated by z = 1 + e l ∆x i + α 2 ∆ 2 x 2i + · · · + α ord(z) ∆ ord(z) x ord(z)i for i even, where ord(z) is the integer such that ord(z)i < n, but (ord(z) + 1)i ≥ n, and α j are certain elements in k.
If z is also an element of the product above, then z = 1 + β 1 e l 1 x p c 1 i l 1 + · · · 1 + β 2 e l 2 x p c 2 i l 2 + · · · · · · , (1 ≤ β r < p, r = 1, . . . , m) for some m, and i r > i 0 if l r = l 0 , i r ≥ i 0 if l r = l 0 . So, the lower degree term of z, as an element of the product must be of the form (β 1 e l 1 +β 2 e l 2 +· · ·+β m e lm )x p a i 0 with p c 1 i 1 = · · · = p cm i m = p a i 0 . But then β 1 e u 1 + β 2 e u 2 + · · · + β m e um = s 0 e l 0 . We have two possible cases according to whether l 0 appears in the factors for z (as an element of a product of H's) or not. In the first case, since the e's are linearly independent, we must have (reordering if necessary) β 1 = s 0 , β 2 = · · · = β m = 0 and l 1 = l 0 . The equality p c 1 i 1 = p a i 0 is contradictory because i 1 > i 0 , and i 0 and i 1 are relatively prime to p. In the second case, we would have linear dependency between the e's.
From here, the result follows.
The next lemmas are direct result of the definition of the integer part function.
Lemma 5. Let a, b be positive integers such that a < b, then the number of multiples of p in the interval ]a, b] is b p − a p .