The Fourier transform on the group $GL_2(R)$ and the action of the overalgebra $gl_4$

We define a kind of 'operational calculus' for $GL_2(R)$. Namely, the group $GL_2(R)$ can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in $R^4$. Therefore the group $GL_4(R)$ acts in $L^2$ on $GL_2(R)$. We transfer the corresponding action of the Lie algebra $gl_4$ to the Plancherel decomposition of $GL_2(R)$, the Lie algebra acts by differential-difference operators with shifts in an imaginary direction. We also write similar formulas for the action of $gl_4\oplus gl_4$ in the Plancherel decomposition of $GL_2(C)$

1 The statement of the paper 1.1. The group GL ( R). Let GL n (R) be the group of invertible real matrices of order n. We denote elements of GL 2 (R) by X = x 11 x 12 x 21 x 22 .
Generators of the Lie algebra gl 2 (R) act by formulas The expressions for the generators do not depend on ε 1 , ε 2 . However the space of the representation depends on ε 1 − ε 2 .
Discrete series. Notice that the group GL 2 (R) acts on the Riemann sphere C = C ∪ ∞ by linear-fractional transformations these transformations preserve the real projective line R ∪ ∞ and therefore preserves its complement C \ R. If det X > 0, then (1.5) leave upper and lower half-planes invariant, if det X < 0, then (1.5) permutes the half-planes. Let n = 1, 2, 3, . . . . Consider the space H n of holomorphic functions ϕ on In fact, ϕ is a pair of holomorphic functions ϕ + and ϕ − determined on halfplanes Im z > 0 and Im z < 0. These functions have boundary values on R in distributional sense (we omit a precise discussion since it is not necessary for our purposes). The space H n is a Hilbert space with respect to the inner product For s ∈ R, δ ∈ Z 2 we define a unitary representation D n,s of GL 2 (R) in H n by D n,s x 11 x 12 In fact, we have operators (1.1) for restricted to the subspace generated by boundary values of functions f + and f − . We denote by Λ discrete the set of all parameters (n, is) of discrete series.
1.2. The Fourier transform. Let ϕ be contained in the space C ∞ 0 (GL 2 (R) of compactly supported function on GL 2 (R). We consider a function that sent each µ = (µ 1 , ε 1 ; µ 2 , ε 2 ) to an operator in C ∞ µ1−µ2,ε1−ε2 given by Next, we define a subset Λ tempered ⊂ Λ by Let us define Plancherel measure dP( µ) on Λ tempered . On the piece Λ principal it is given by On n-th piece of Λ discrete it is given by dP = n 8π 3 ds.
Consider the space L 2 of functions Q on Λ tempered taking values in the space of Hilbert-Schmidt operators in the corresponding Hilbert spaces and satisfying the condition This is a Hilbert space with respect to the inner product According the Plancherel theorem, for any F 1 , F 2 ∈ C ∞ 0 (GL 2 (R)) we have Moreover, the map F → T (F ) extends to a unitary operator L 2 (GL 2 (R)) → L.
1.3. Overgroup. Let Mat 2 (R) be the space of all real matrices of order 2. By Gr 4,2 (R) we denote the Grassmannian of 2-dimensional subspaces in R 2 ⊕R 2 . For any operator R 2 → R 2 its graph is an element Gr 4,2 (R). The set Mat 2 (R) of such operators is an open dense chart in Gr 4,2 (R).
The group GL 4 (R) acts in a natural way in R 4 and therefore on the Grassmannian. In the chart Mat 2 (R) the action is given by the formula (see, e.g., [18], Theorem 2.3.2) where A B C D is an element of GL 4 (R) written as a block matrix of size 2 + 2.
The Jacobian of this transformation is (see, e.g., [18], Theorem 2.3.2) For σ ∈ iR we define a unitary representation of GL 4 (R) in L 2 (Mat 2 (R)) by (these representations are contained in degenerate principal series).
The group GL 2 (R) is an open dense subset in Mat 2 (R). Therefore, we can identify the spaces L 2 on GL 2 (R) and Mat 2 (R). For this we consider a unitary operator J : Formula (1.6) extends this formula to the whole group GL 4 (R). The Lie algebra gl 4 acts in the space of functions on Mat 2 (R) by first order differential operators, which can be easily written; a list of formulas for all generators e kl , where k, l = 1, 2, 3, 4, is given below in Subs. (2.5). We restrict this action to the space of smooth compactly supported functions on GL 2 (R). Notice that the operators i · e kl are symmetric on this domain, but some of them are not essentially selfadjoint.
Our purpose is to write explicitly the images E kl of operators e kl under the Fourier transform.

Formulas.
Operators T µ1,ε1;µ2;ε2 (F ) have the form (1.7) Recall that functions K are holomorphic in µ 1 , µ 2 . We wish to write operators E kl on kernels K. The complete list is contained below in Subsect. 2.5, here we present two basic expressions.
The algebra gl 4 (R) can be decomposed as a linear space into a direct sum of four subalgebras a, b, c, d consisting of matrices of the form * 0 0 0 , The subalgebras a and d are isomorphic to gl 2 (R), subalgebras b and c are Abelian.
Formulas for the action of a and d immediately follow from the definition of the Fourier transform. To obtain formulas for the whole gl 4 , it is sufficient to write expressions for one generator of b, say E 14 , and one generator of c, say E 32 . After this other generators can be obtained by evaluation of commutators.
We define shift operators In this notation, 1.5. Remarks on a general problem. In [17] the author formulated the following question: Assume that we know the explicit Plancherel formula for the restriction of a unitary representation ρ of a group G to a subgroup H. Is it possible to write the action of the Lie algebra of G in the direct integral of representations of H?
Now it seems that an answer to this question is affirmative. The initial paper [17] contains a solution for a tensor product 3 of a highest and lowest weight representations of SL 2 (R). In this case the overalgebra acts by differential-difference operators in the space L 2 (S 1 × R + ) having the form where D j are second order differential operators in the variable ϕ.
In [10]- [13] Molchanov solved several rank 1 problems of this type, expressions are similar, but there appear differential operators of order 4. In [20] the action of the overalgebra in restrictions from GL n+1 (C) to GL n (C), in this case differential operators have order n. In all the cases examined by now shift operators in the imaginary direction are present.
In the present paper, we write the action of the overalgebra in the restriction of a degenerate principal series of the group GL 4 (R) to GL 2 (R). Notice that canonical overgroups exist for all 10 series of real classical groups 4 . Moreover overgroups exist for all 52 series of classical semisimple symmetric spaces G/H, see [9], [15], see also [18], Addendum D.6. So the problem makes sense for all classical symmetric spaces.
Shturm-Liouville problems for difference operators in L 2 (R) in the imaginary direction arise in a natural way in the theory of hypergeometric orthogonal polynomials, see, e.g., [1], [8], apparently a first example (the Meixner-Pollaszek system) was discovered by J. Meixner in 1930s. On such operators with continuous spectra see [16], [6], [19]. See also a multi-rank work of I. Cherednik [2] on Harish-Chandra spherical transforms.
1.6. The Fourier transform on GL 2 (C). For a detailed exposition of representations of the Lorentz group, see [14]. For ν, ν ′ ∈ C satisfying ν −ν ′ ∈ Z we define the function z ν ν ′ on the multiplicative group of C by We consider the space of C ∞ -smooth functions ϕ on C such that is C ∞ -smooth at 0. These representations form the (nonunitary) principal series.
It is convenient to complexify the Lie algebra of GL 2 (C), Under this isomorphism, the operators of the Lie algebra act in our representation by Formally, we have duplicated expressions (1.2)-(1.3).
For any compactly supported smooth function F on GL 2 (C) we define its Fourier transform as an operator-valued function on ∆ given by The Plancherel formula is the following identity 1.7. Overgroup for GL 2 (C). Consider the complex Grassmannian Gr 4,2 (C) of 2-dimensional planes in C 4 , again the set Mat 2 (C) is an open dense set on Gr 4,2 (C). For σ, σ ′ ∈ iR consider a unitary representation R σ,σ ′ of GL 4 (C) in L 2 Mat 2 (C) given by We define a unitary operator J : L 2 Mat 2 (C) → L 2 Mat 2 (C) by In this way we get a unitary representation U σ, 1.8. Formulas for GL 2 (C). We wish to write the action of the Lie algebra in the Plancherel decomposition of GL 2 (C). Denote the standard generators of gl 4 (C) ⊕ 0 and 0 ⊕ gl 4 (C) by E kl and E kl respectively. Define the following shift operators and similar operators V 2 and V ′ 2 shifting µ 2 and µ ′ 2 . Then

Calculations
2.1. The expression for kernel.
Lemma 2.1 The kernel K(·) of an integral operator T µ1,ε1;µ2,ε2 (F ) is given by the formula (2.1) For F ∈ C ∞ 0 GL 2 (R) the integration is actually taken over a bounded domain.

Proof. By the definition
In the interior integral, we pass from the variables x 11 , x 12 , x 21 , x 22 to new variables u, v, w, s defined by The Jacobi matrix of this transformation is triangular, and the Jacobian is |u|. The inverse transformation is We also have x 11 x 22 − x 12 x 21 = uw.
After the change of variables we come to where K(·) is given by (2.1).
A function F has a compact support in R 4 \ {x 11 x 22 − x 12 x 21 = 0}. So, actually, x 21 = v, x 11 = u − tv, x 22 = w + sv are contained in a bounded domain. This implies the second claim of the lemma.

A verification of the formula for E 14 .
It is easy to verify that the operator e 14 in C ∞ 0 GL 2 (R) is given by Therefore, (the integration is taken over R 3 on default). We must verify that (2.8) coincides with Below we establish two formulas Considering the sum of (2.10) and (2.11) with coefficients −1/2−σ+µ1 µ1−µ2 we get coincidence of (2.8) and (2.9); for this, we use he identities Now let us check (2.10). The following identity can be verified by a straightforward calculation (with (2.6) and (2.5)): Therefore the left-hand side of (2.10) equals to We integrate this expression by parts in the variable w and come to (2.10).
To check (2.11), we verify the identity and after this integrate by parts as above.

A verification of the formula for E 32 .
We have Therefore, On the other hand, We must verify that (2.12) and (2.12) are equal. As in the previous subsection, this statement is reduced to a pair of identities Let us verify (2.14). It can be easily checked (with (2.7), (2.4), (2.5)) that We substitute this to the left-hand side of (2.14) and come to Integrating by parts, we get After a summation we come to the right-hand side of (2.14). A proof of (2.15) is similar, we use the identity and repeat the same steps.  Denote by E kl the corresponding operators E kl on kernels K. Formulas for operators of groups a), b) immediately follow from the definition of the Fourier transform,
2.6. The case of GL 2 (C). Notice that formulas in Subsections 1.1-1.4 for SL 2 (R) and in Subsections 1.6-1.8 are very similar, except the Plancherel formulas.
The analog of formula (2.1) is Its derivation is based on the same change of variables (2.2), its real Jacobian is uu. A further calculation one-to-one follows the calculation for GL 2 (R).