The Fourier Transform on the Group GL2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_2({\mathbb R })$$\end{document} and the Action of the Overalgebra gl4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {gl}_4$$\end{document}

We define a kind of ’operational calculus’ for the Fourier transform on the group GL2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_2({\mathbb R })$$\end{document}. Namely, GL2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_2({\mathbb R })$$\end{document} can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R }^4$$\end{document}. Therefore the group GL4(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_4({\mathbb R })$$\end{document} acts in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} on GL2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_2({\mathbb R })$$\end{document}. We transfer the corresponding action of the Lie algebra gl4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {gl}_4$$\end{document} to the Plancherel decomposition of GL2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_2({\mathbb R })$$\end{document}, the algebra acts by differential-difference operators with shifts in an imaginary direction. We also write similar formulas for the action of gl4⊕gl4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {gl}_4\oplus \mathfrak {gl}_4$$\end{document} in the Plancherel decomposition of GL2(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_2({\mathbb C })$$\end{document}.

Generators of the Lie algebra gl 2 (R) act by formulas 3) 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 C \ R satisfying In fact, ϕ is a pair of holomorphic functions ϕ + and ϕ − determined on half-planes 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 x 21 x 22 In fact, we have operators (1.1) for restricted to the subspace generated by boundary values of functions f + and f − . We denote by discrete the set of all parameters of discrete series.
Next, we define a subset tempered ⊂ by tempered := principal ∪ discrete .
Let us define Plancherel measure dP( μ) on tempered . On the piece principal it is given by On n-th piece of discrete it is given by dP = n 8π 3 ds.
Consider the space L 2 of functions Q on 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 , Moreover, the map F → T (F) extends to a unitary operator L 2 (GL 2 (R)) → L 2 .

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)

A B C D
: 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 (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 given by For this subgroup we get the usual left-right action in L 2 GL 2 (R) , 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 Sect. 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 self-adjoint. Our purpose is to write explicitly the images E kl of operators e kl under the Fourier transform.

Formulas
(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 Sect. 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 The operators E 14 , E 32 are given by the formulas Remark We emphasize that these formulas determining unbounded skew-symmetric operators in L include shifts transversal to the contour of integration in the Plancherel formula (the contour corresponds to pure imaginary μ 1 , μ 2 and (1.8), (1.9) are a shift operators in real directions).

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 2 of a highest and lowest weight representations of SL 2 (R). In this case the overalgebra acts by differentialdifference operators in the space where D 1 , D 2 , D 3 are differential operators in the variable ϕ of orders 0, 1, 2 respectively.
In [9][10][11][12][13] Molchanov solved several rank 1 problems of this type, expressions are similar, but there appear differential operators of order 4. In [20] there was obtained 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 formulas include shift operators in the imaginary direction.
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. 3 Moreover overgroups exist for all 52 series of classical semisimple symmetric spaces G/H , see [8,15], see also [18], Addendum D.6. So the problem makes sense for all classical symmetric spaces.
Sturm-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., [1,7], apparently a first example (the Meixner-Pollaczek system) was discovered by J. Meixner in 1930s. On such operators with continuous spectra see [5,16,19]. See also a multi-rank work of Cherednik [2] on Harish-Chandra spherical transforms.

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 z ν ν := z ν z ν := |z| 2ν z ν −ν .
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 then the representation T μ 1 ,μ 1 ;μ 2 ,μ 2 is unitary in L 2 . Denote by tempered the set of such tuples (μ 1 , μ 1 ; μ 2 , μ 2 ), it is a union of a countable family of parallel 2-dimensional real planes in C 4 , we equip it by a natural Lebesgue measure dλ(μ). 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 where C is an explicit constant. Denote by K (t, s|μ 1 , μ 1 ; μ 2 , μ 2 ) the kernel of the operator T μ 1 ,μ 1 ;μ 2 ,μ 2 (F),

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 .

Theorem 1.2
The operators E 14 , E 14 , E 23 , E 23 are given by the formulas
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 +μ 1 μ 1 −μ 2 and −1/2−σ +μ 2 μ 1 −μ 2 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, where On the other hand, We must verify that (2.12) and (2.13) are equal. As in the previous subsection, this statement is reduced to a pair of identities (2.14) (2.15) 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.

Table of Formulas
First, we present formulas for the action of the Lie algebra gl 4 corresponding to U σ , see (1.6). Denote generators of gl 4 by e kl , where 1 k, l 4. Denote by ∂ pq the partial derivatives ∂ ∂ x pq , where p, q = 1, 2. The generators e kl naturally split into 4 groups corresponding to blocks A, B, C, D in (1.6).
(a) Generators corresponding to the block A form a Lie algebra gl 2 : (b) Generators corresponding to the block D also form a Lie algebra gl 2 : (c) Elements corresponding to the block B form a 4-dimensional Abelian Lie algebra: 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, Next, and

The Case of GL 2 (C)
Notice that formulas in Sects. 1.1-1.4 for SL 2 (R) and in Sects. 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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.