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

TL;DR

This paper develops an operational calculus for the group $GL_2(R)$ by leveraging its embedding in the Grassmannian and explores the action of the Lie algebra $gl_4$ on its Plancherel decomposition, involving differential-difference operators.

Contribution

It introduces a novel approach to analyze $GL_2(R)$ using the action of $gl_4$ on its harmonic analysis decomposition, connecting group actions with differential-difference operators.

Findings

Derived explicit formulas for $gl_4$ action on $GL_2(R)$

Expressed the Lie algebra action as differential-difference operators

Extended the analysis to $GL_2(C)$ with $gl_4 imes gl_4$ action

Abstract

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 $GL_2(C)$

Equations

$$X=\left(\begin{array}{cc}{x}_{11}& x_{12}\\ x_{21}& x_{22}\end{array}\right).$$

$$(\det X{)}^{-2}dX=(\det X{)}^{-2}d{x}_{11}d{x}_{12}d{x}_{21}d{x}_{22}.$$

$$x^{\mu //\epsilon }:=|x{|}^{\mu }\mathrm{sgn}(x{)}^{\epsilon }.$$

$$({\mu }_1,{\epsilon }_1;{\mu }_2,{\epsilon }_2),$$

$$\Lambda \simeq \mathbb{C}\times {\mathbb{Z}}_2\times \mathbb{C}\times {\mathbb{Z}}_2.$$

$$T_{\mu_{1},\varepsilon_{1};\mu_{2},\varepsilon_{2}}\begin{pmatrix}x_{11}&x_{12}\\ x_{21}&x_{22}\end{pmatrix}\varphi(t)=\\ =\varphi\Bigl{(}\frac{x_{12}+tx_{22}}{x_{11}+tx_{21}}\Bigr{)}\cdot(x_{11}+tx_{21})^{-1+\mu_{1}-\mu_{2}/\!\!/\varepsilon_{1}-\varepsilon_{2}}\det\begin{pmatrix}x_{11}&x_{12}\\ x_{21}&x_{22}\end{pmatrix}^{1/2+\mu_{2}/\!\!/\varepsilon_{2}}.$$

$$\phi (-1/t)(-t{)}^{-1+{\mu }_1-{\mu }_2//{\epsilon }_1-{\epsilon }_2}$$

$$L_{11}$$

$$L_{21}$$

$$A_{{\mu }_1,{\epsilon }_1;{\mu }_2,{\epsilon }_2}:C_{{\mu }_1-{\mu }_2,{\epsilon }_1-{\epsilon }_2}^{\infty }\to C_{{\mu }_2-{\mu }_1,{\epsilon }_1-{\epsilon }_2}^{\infty }$$

$$A_{{\mu }_1,{\epsilon }_1;{\mu }_2,{\epsilon }_2}f(t):={\int }_{\mathbb{R}}(t-s{)}^{-1-{\mu }_1+{\mu }_2//{\epsilon }_1-{\epsilon }_2}f(s)ds.$$

$$A_{{\mu }_1,{\epsilon }_1;{\mu }_2,{\epsilon }_2}{T}_{{\mu }_1,{\epsilon }_1;{\mu }_2,{\epsilon }_2}=T_{{\mu }_2,{\epsilon }_2;{\mu }_1,{\epsilon }_1}{A}_{{\mu }_1,{\epsilon }_1;{\mu }_2,{\epsilon }_2}$$

$$X:z\mapsto \frac{x_{12}+z{x}_{22}}{x_{11}+z{x}_{21}},$$

$${\int }_{\mathbb{C}\setminus \mathbb{R}}|\phi (z){|}^2|\mathrm{Im}z{|}^{-n-1}d\mathrm{Re}zd\mathrm{Im}z<\infty .$$

$$⟨{\phi }_1,{\phi }_2⟩={\int }_{\mathbb{C}\setminus \mathbb{R}}{\phi }_1(z)\overline{{\phi }_2(z)}|\mathrm{Im}z{|}^{-n-1}d\mathrm{Re}zd\mathrm{Im}z.$$

$$D_{n,s}\begin{pmatrix}x_{11}&x_{12}\\ x_{21}&x_{22}\end{pmatrix}\varphi(z)=\varphi\Bigl{(}\frac{x_{12}+zx_{22}}{x_{11}+zx_{21}}\Bigr{)}(x_{11}+zx_{21})^{-1-n}\times\\ \times\det\begin{pmatrix}x_{11}&x_{12}\\ x_{21}&x_{22}\end{pmatrix}^{1/2+n/2+is/\!\!/\delta}.$$

$${\mu }_1=-n/2-is,{\epsilon }_1=-1+n+\delta ;{\mu }_2=n/2+is,{\epsilon }_2=\delta$$

$$T_{{\mu }_1,{\epsilon }_1;{\mu }_2,{\epsilon }_2}(F)={\int }_{{\mathrm{GL}}_2(\mathbb{R})}F(X)T_{{\mu }_1,{\epsilon }_1;{\mu }_2,{\epsilon }_2}(X)\frac{dX}{\det (X{)}^2}.$$

$${\Lambda }_{\mathrm{tempered}}:={\Lambda }_{\mathrm{principal}}\cup {\Lambda }_{\mathrm{discrete}}.$$

$$d\mathcal{P}(i{s}_1,0;i{s}_2,{\epsilon }_2)=\frac{1}{16{\pi }^3}(s_1-s_2)\tanh \pi (s_1-s_2)/2d{s}_{1}d{s}_2;$$

$$d\mathcal{P}(i{s}_1,1;i{s}_2,{\epsilon }_2)=\frac{1}{16{\pi }^3}(s_1-s_2)\coth \pi (s_1-s_2)/2d{s}_{1}d{s}_2.$$

$$d\mathcal{P}=\frac{n}{8{\pi }^3}ds.$$

$$\int_{\Lambda_{\mathrm{tempered}}}\mathop{\mathrm{tr}}\nolimits\Bigl{(}Q(\widetilde{\mu})Q^{*}(\widetilde{\mu})\Bigr{)}\,d\mathcal{P}(\widetilde{\mu})<\infty.$$

$$\langle Q_{1},Q_{2}\rangle_{\mathcal{L}^{2}}:=\int_{\Lambda_{\mathrm{tempered}}}\mathop{\mathrm{tr}}\nolimits\Bigl{(}Q_{1}(\widetilde{\mu})Q_{2}^{*}(\widetilde{\mu})\Bigr{)}\,\,d\mathcal{P}(\widetilde{\mu}).$$

$$⟨F_1,F_2{⟩}_{L^2({\mathrm{GL}}_2(\mathbb{R}))}=⟨T(F_1),T(F_2){⟩}_{{\mathcal{L}}^2({\Lambda }_{\mathrm{tempered}})}.$$

$$\left(\begin{array}{cc}A& B\\ C& D\end{array}\right):X\mapsto (A+XC{)}^{-1}(B+XD),$$

$$\det (A+XC{)}^{-4}\det {\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)}^2.$$

$$R_{\sigma}\begin{pmatrix}A&B\\ C&D\end{pmatrix}F(X)=F\bigl{(}(A+XC)^{-1}(B+XD)\bigr{)}\times\\ \times\,|\det(A+XC)|^{-2+2\sigma}\left|\det\begin{pmatrix}A&B\\ C&D\end{pmatrix}\right|^{1-\sigma}$$

$$J:\,L^{2}\bigl{(}\mathrm{Mat}_{2}({\mathbb{R}})\bigr{)}\to L^{2}\bigl{(}\mathrm{GL}_{2}({\mathbb{R}})\bigr{)}$$

$$J_{\sigma }F(X)=F(X)\cdot |\det X{|}^{1-\sigma }$$

$$U_{\sigma}\begin{pmatrix}A&B\\ C&D\end{pmatrix}F(X)=F\bigl{(}(A+XC)^{-1}(B+XD)\bigr{)}\times\\ \times\Biggl{|}\frac{\det(A+XC)\det(B+XD)}{\det X\det\begin{pmatrix}A&B\\ C&D\end{pmatrix}}\Biggr{|}^{-1+\sigma}.$$

Full text

The Fourier transform on the group $\mathrm{GL}_{2}({\mathbb{R}})$ and the action of the overalgebra $\mathfrak{gl}_{4}$

Yu.A.Neretin111Supported by the grant FWF, Project P28421.

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

1 The statement of the paper

1.1. The group $\mathrm{GL}_{(}{\mathbb{R}})$. Let $\mathrm{GL}_{n}({\mathbb{R}})$ be the group of invertible real matrices of order $n$. We denote elements of $\mathrm{GL}_{2}({\mathbb{R}})$ by

[TABLE]

The Haar measure on $\mathrm{GL}_{2}({\mathbb{R}})$ is given by

[TABLE]

Recall some basic facts on representations of the group $\mathrm{GL}_{2}({\mathbb{R}})$, for systematic exposition of the representation theory of $\mathrm{SL}_{2}({\mathbb{R}})$, see, e.g., [5], [7]. Denote by ${\mathbb{R}}^{\times}$ the multiplicative group of ${\mathbb{R}}$. Let $\mu\in{\mathbb{C}}$ and $\varepsilon\in{\mathbb{Z}}/2{\mathbb{Z}}$. We define the function $x^{\mu/\!\!/\varepsilon}$ on ${\mathbb{R}}^{\times}$ by

[TABLE]

Denote by $\Lambda$ the set of all collections

[TABLE]

i.e.,

[TABLE]

For each element of $\Lambda$ we define a representation $T_{\mu,\varepsilon}$ of $\mathrm{GL}_{2}({\mathbb{R}})$ in the space of functions on ${\mathbb{R}}$ by

[TABLE]

A space of functions can be specified in various ways, it is convenient to consider the space $C^{\infty}_{\mu_{1}-\mu_{2},\varepsilon_{1}-\varepsilon_{2}}$ of $C^{\infty}$-functions on ${\mathbb{R}}$ such that222This condition means that functions $\varphi$ are smooth as sections of line bundles on the projective line ${\mathbb{R}}\cup\infty$.

[TABLE]

also is $C^{\infty}$-smooth.

Thus, for any fixed $X\in\mathrm{GL}_{2}({\mathbb{R}})$ we get an operator-valued function $(\mu_{1},\varepsilon_{1};\mu_{2},\varepsilon_{2})\mapsto T_{\mu_{1},\varepsilon_{1};\mu_{2},\varepsilon_{2}}$ on $\Lambda$ holomorphic in the variables $\mu_{1}$, $\mu_{2}$.

Generators of the Lie algebra $\mathfrak{g}\mathfrak{l}_{2}({\mathbb{R}})$ act by formulas

[TABLE]

The expressions for the generators do not depend on $\varepsilon_{1}$, $\varepsilon_{2}$. However the space of the representation depends on $\varepsilon_{1}-\varepsilon_{2}$.

Consider integral operators

[TABLE]

defined by

[TABLE]

The integral is convergent if $\mathop{\mathrm{Re}}\nolimits(\mu_{1}-\mu_{2})<0$ and determines a function holomorphic in $\mu_{1}$, $\mu_{2}$. As usual (see, e.g., [4], §I.3), the integral admits a meromorphic continuation to the whole plane $(\mu_{1},\mu_{2})\in{\mathbb{C}}^{2}$ with poles at $\mu_{1}-\mu_{2}=0$, $1$, $2$, …. The operators $A_{\mu_{1},\varepsilon_{1};\mu_{2},\varepsilon_{2}}$ are intertwinig,

[TABLE]

(parameters $(\mu_{1},\varepsilon_{1})$ and $(\mu_{2},\varepsilon_{2})$ of $T$ are transposed). If $\mu_{1}-\mu_{2}\neq\pm 1$, $\pm 2$, …, then representations $T_{\mu_{1},\varepsilon_{1};\mu_{2},\varepsilon_{2}}$ are irreducible and operators $T_{\mu_{1},\varepsilon_{1};\mu_{2},\varepsilon_{2}}$ are invertible.

Representations $T_{\mu_{1},\varepsilon_{1};\mu_{2},\varepsilon_{2}}$ form a so-called (nonunitary) principal series of representations. Recall description of some unitary irreducible representations of $\mathrm{GL}_{2}({\mathbb{R}})$.

Unitary principal series. If $\mu_{1}$, $\mu_{2}\in i{\mathbb{R}}$, then the representation is unitary in $L^{2}({\mathbb{R}})$. Denote by $\Lambda_{\mathrm{principal}}\subset\Lambda$ the subset consisting of tuples $(is_{1},\varepsilon_{1};is_{2},\varepsilon_{2})$ with $s_{1}\geqslant s_{2}$.

Discrete series. Notice that the group $\mathrm{GL}_{2}({\mathbb{R}})$ acts on the Riemann sphere $\overline{C}={\mathbb{C}}\cup\infty$ by linear-fractional transformations

[TABLE]

these transformations preserve the real projective line ${\mathbb{R}}\cup\infty$ and therefore preserves its complement ${\mathbb{C}}\setminus{\mathbb{R}}$. 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 $\varphi$ on ${\mathbb{C}}\setminus{\mathbb{R}}$ satisfying

[TABLE]

In fact, $\varphi$ is a pair of holomorphic functions $\varphi_{+}$ and $\varphi_{-}$ determined on half-planes $\mathop{\mathrm{Im}}\nolimits z>0$ and $\mathop{\mathrm{Im}}\nolimits z<0$. These functions have boundary values on ${\mathbb{R}}$ 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

[TABLE]

For $s\in{\mathbb{R}}$, $\delta\in{\mathbb{Z}}_{2}$ we define a unitary representation $D_{n,s}$ of $\mathrm{GL}_{2}({\mathbb{R}})$ in $H_{n}$ by

[TABLE]

In fact, we have operators (1.1) for

[TABLE]

restricted to the subspace generated by boundary values of functions $f_{+}$ and $f_{-}$.

We denote by $\Lambda_{\mathrm{discrete}}$ the set of all parameters $(n,is)$ of discrete series.

1.2. The Fourier transform. Let $\varphi$ be contained in the space $C_{0}^{\infty}(\mathrm{GL}_{2}({\mathbb{R}})$ of compactly supported function on $\mathrm{GL}_{2}({\mathbb{R}})$. We consider a function that sent each $\widetilde{\mu}=(\mu_{1},\varepsilon_{1};\mu_{2},\varepsilon_{2})$ to an operator in $C^{\infty}_{\mu_{1}-\mu_{2},\varepsilon_{1}-\varepsilon_{2}}$ given by

[TABLE]

Next, we define a subset $\Lambda_{\mathrm{tempered}}\subset\Lambda$ by

[TABLE]

Let us define Plancherel measure $d\mathcal{P}(\widetilde{\mu})$ on $\Lambda_{\mathrm{tempered}}$. On the piece $\Lambda_{\mathrm{principal}}$ it is given by

[TABLE]

On $n$-th piece of $\Lambda_{\mathrm{discrete}}$ it is given by

[TABLE]

Consider the space $\mathcal{L}^{2}$ of functions $Q$ on $\Lambda_{\mathrm{tempered}}$ taking values in the space of Hilbert–Schmidt operators in the corresponding Hilbert spaces and satisfying the condition

[TABLE]

This is a Hilbert space with respect to the inner product

[TABLE]

According the Plancherel theorem, for any $F_{1}$, $F_{2}\in C^{\infty}_{0}(\mathrm{GL}_{2}({\mathbb{R}}))$ we have

[TABLE]

Moreover, the map $F\to T(F)$ extends to a unitary operator $L^{2}(\mathrm{GL}_{2}({\mathbb{R}}))\to\mathcal{L}$.

1.3. Overgroup. Let $\mathrm{Mat}_{2}({\mathbb{R}})$ be the space of all real matrices of order 2. By $\mathrm{Gr}_{4,2}({\mathbb{R}})$ we denote the Grassmannian of 2-dimensional subspaces in ${\mathbb{R}}^{2}\oplus{\mathbb{R}}^{2}$. For any operator ${\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ its graph is an element $\mathrm{Gr}_{4,2}({\mathbb{R}})$. The set $\mathrm{Mat}_{2}({\mathbb{R}})$ of such operators is an open dense chart in $\mathrm{Gr}_{4,2}({\mathbb{R}})$.

The group $\mathrm{GL}_{4}({\mathbb{R}})$ acts in a natural way in ${\mathbb{R}}^{4}$ and therefore on the Grassmannian. In the chart $\mathrm{Mat}_{2}({\mathbb{R}})$ the action is given by the formula (see, e.g., [18], Theorem 2.3.2)

[TABLE]

where $\begin{pmatrix}A&B\\ C&D\end{pmatrix}$ is an element of $\mathrm{GL}_{4}({\mathbb{R}})$ written as a block matrix of size $2+2$. The Jacobian of this transformation is (see, e.g., [18], Theorem 2.3.2)

[TABLE]

For $\sigma\in i{\mathbb{R}}$ we define a unitary representation of $\mathrm{GL}_{4}({\mathbb{R}})$ in $L^{2}(\mathrm{Mat}_{2}({\mathbb{R}}))$ by

[TABLE]

(these representations are contained in degenerate principal series).

The group $\mathrm{GL}_{2}({\mathbb{R}})$ is an open dense subset in $\mathrm{Mat}_{2}({\mathbb{R}})$. Therefore, we can identify the spaces $L^{2}$ on $\mathrm{GL}_{2}({\mathbb{R}})$ and $\mathrm{Mat}_{2}({\mathbb{R}})$. For this we consider a unitary operator

[TABLE]

given by

[TABLE]

This determines a unitary representation $U_{\sigma}:=J_{\sigma}R_{\sigma}J_{\sigma}^{-1}$ of $\mathrm{GL}_{4}({\mathbb{R}})$ in L^{2}\bigl{(}\mathrm{GL}_{2}({\mathbb{R}})\bigr{)}, the explicit expression is

[TABLE]

Consider a subgroup $\mathrm{GL}_{2}({\mathbb{R}})\times\mathrm{GL}_{2}({\mathbb{R}})\subset\mathrm{GL}_{4}({\mathbb{R}})$ consisting of matrices $\begin{pmatrix}A&0\\ 0&D\end{pmatrix}$. For this subgroup we get the usual left-right action in L^{2}\bigl{(}\mathrm{GL}_{2}({\mathbb{R}})\bigr{)},

[TABLE]

Formula (1.6) extends this formula to the whole group $\mathrm{GL}_{4}({\mathbb{R}})$. The Lie algebra $\mathfrak{g}\mathfrak{l}_{4}$ acts in the space of functions on $\mathrm{Mat}_{2}({\mathbb{R}})$ 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). We restrict this action to the space of smooth compactly supported functions on $\mathrm{GL}_{2}({\mathbb{R}})$. Notice that the operators $i\cdot e_{kl}$ 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.

1.4. Formulas. Operators $T_{\mu_{1},\varepsilon_{1};\mu_{2};\varepsilon_{2}}(F)$ have the form

[TABLE]

Recall that functions $K$ are holomorphic in $\mu_{1}$, $\mu_{2}$.

We wish to write operators $E_{kl}$ on kernels $K$. The complete list is contained below in Subsect. 2, here we present two basic expressions.

The algebra $\mathfrak{gl}_{4}({\mathbb{R}})$ can be decomposed as a linear space into a direct sum of four subalgebras $\mathfrak{a}$, $\mathfrak{b}$, $\mathfrak{c}$, $\mathfrak{d}$ consisting of matrices of the form

[TABLE]

The subalgebras $\mathfrak{a}$ and $\mathfrak{d}$ are isomorphic to $\mathfrak{gl}_{2}({\mathbb{R}})$, subalgebras $\mathfrak{b}$ and $\mathfrak{c}$ are Abelian.

Formulas for the action of $\mathfrak{a}$ and $\mathfrak{d}$ immediately follow from the definition of the Fourier transform. To obtain formulas for the whole $\mathfrak{gl}_{4}$, it is sufficient to write expressions for one generator of $\mathfrak{b}$, say $E_{14}$, and one generator of $\mathfrak{c}$, say $E_{32}$. After this other generators can be obtained by evaluation of commutators.

We define shift operators $V_{1}^{+}$, $V_{1}^{-}$, $V_{2}^{+}$, $V_{2}^{-}$ by

[TABLE]

In this notation,

[TABLE]

[TABLE]

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 $\rho$ 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 product333The problem of a decomposition of a tensor product $\rho_{1}\otimes\rho_{2}$ of representations of a group $G$ is a special case of a decomposition of restrictions. Indeed, $\rho_{1}\otimes\rho_{2}$ is a representation of the group $G\times G$, and we must restrict this representation to the diagonal $G$. of a highest and lowest weight representations of $\mathrm{SL}_{2}({\mathbb{R}})$. In this case the overalgebra acts by differential-difference operators in the space $L^{2}(S^{1}\times{\mathbb{R}}_{+})$ having the form

[TABLE]

where $D_{j}$ are second order differential operators in the variable $\varphi$.

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 $\mathrm{GL}_{n+1}({\mathbb{C}})$ to $\mathrm{GL}_{n}({\mathbb{C}})$, 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 $\mathrm{GL}_{4}({\mathbb{R}})$ to $\mathrm{GL}_{2}({\mathbb{R}})$. Notice that canonical overgroups exist for all 10 series of real classical groups444Emphasize that in the present paper we consider groups $\mathrm{GL}$ and not $\mathrm{SL}$, also we must consider groups $\mathrm{U}(p,q)$ and not $\mathrm{SU}(p,q)$.. 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 $L^{2}({\mathbb{R}})$ in the imaginary direction

[TABLE]

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 $\mathrm{GL}_{2}({\mathbb{C}})$. For a detailed exposition of representations of the Lorentz group, see [14]. For $\nu$, $\nu^{\prime}\in{\mathbb{C}}$ satisfying $\nu-\nu^{\prime}\in{\mathbb{Z}}$ we define the function $z^{\nu\|\nu^{\prime}}$ on the multiplicative group of ${\mathbb{C}}$ by

[TABLE]

Denote by $\Delta$ the set of all $(\mu_{1},\mu_{1}^{\prime};\mu_{2},\mu_{2}^{\prime})\in{\mathbb{C}}^{4}$ such that $\mu_{1}-\mu_{1}^{\prime}\in{\mathbb{Z}}$, $\mu_{2}-\mu_{2}^{\prime}\in{\mathbb{Z}}$. For $(\mu_{1},\mu_{1}^{\prime};\mu_{2},\mu_{2}^{\prime})\in\Delta$ we define a representation of $\mathrm{GL}_{2}({\mathbb{C}})$ in the space of functions on ${\mathbb{C}}$ by

[TABLE]

We consider the space of $C^{\infty}$-smooth functions $\varphi$ on ${\mathbb{C}}$ such that

[TABLE]

is $C^{\infty}$-smooth at 0. These representations form the (nonunitary) principal series.

It is convenient to complexify the Lie algebra of $\mathrm{GL}_{2}({\mathbb{C}})$,

[TABLE]

Under this isomorphism, the operators of the Lie algebra act in our representation by

[TABLE]

Formally, we have duplicated expressions (1.2)–(1.3).

If

[TABLE]

then the representation $T_{\mu_{1},\mu_{1}^{\prime};\mu_{2},\mu_{2}^{\prime}}$ is unitary in $L^{2}$. Denote by $\Delta_{\mathrm{tempered}}$ the set of such tuples $(\mu_{1},\mu_{1}^{\prime};\mu_{2},\mu_{2}^{\prime})$, it is a union of a countable family of parallel 2-dimensional real planes in ${\mathbb{C}}^{4}$, we equip it by a natural Lebesgue measure $d\lambda(\mu)$.

For any compactly supported smooth function $F$ on $\mathrm{GL}_{2}({\mathbb{C}})$ we define its Fourier transform as an operator-valued function on $\Delta$ given by

[TABLE]

The Plancherel formula is the following identity

[TABLE]

where $C$ is an explicit constant.

Denote by $K(t,s|\mu_{1},\mu_{1}^{\prime};\mu_{2},\mu_{2}^{\prime})$ the kernel of the operator $T_{\mu_{1},\mu_{1}^{\prime};\mu_{2},\mu_{2}^{\prime}}(F)$,

[TABLE]

1.7. Overgroup for $\mathrm{GL}_{2}({\mathbb{C}})$. Consider the complex Grassmannian $\mathrm{Gr}_{4,2}({\mathbb{C}})$ of 2-dimensional planes in ${\mathbb{C}}^{4}$, again the set $\mathrm{Mat}_{2}({\mathbb{C}})$ is an open dense set on $\mathrm{Gr}_{4,2}({\mathbb{C}})$. For $\sigma$, $\sigma^{\prime}\in i{\mathbb{R}}$ consider a unitary representation $R_{\sigma,\sigma^{\prime}}$ of $\mathrm{GL}_{4}({\mathbb{C}})$ in L^{2}\bigl{(}\mathrm{Mat}_{2}({\mathbb{C}})\bigr{)} given by

[TABLE]

We define a unitary operator J:L^{2}\bigl{(}\mathrm{Mat}_{2}({\mathbb{C}})\bigr{)}\to L^{2}\bigl{(}\mathrm{Mat}_{2}({\mathbb{C}})\bigr{)} by

[TABLE]

In this way we get a unitary representation $U_{\sigma,\sigma^{\prime}}=J_{\sigma,\sigma^{\prime}}R_{\sigma,\sigma^{\prime}}J_{\sigma,\sigma^{\prime}}^{-1}$ of $\mathrm{GL}_{4}({\mathbb{C}})$ in L^{2}\bigl{(}\mathrm{Mat}_{2}({\mathbb{C}})\bigr{)}:

[TABLE]

1.8. Formulas for $\mathrm{GL}_{2}({\mathbb{C}})$. We wish to write the action of the Lie algebra

[TABLE]

in the Plancherel decomposition of $\mathrm{GL}_{2}({\mathbb{C}})$. Denote the standard generators of $\mathfrak{gl}_{4}({\mathbb{C}})\oplus 0$ and $0\oplus\mathfrak{gl}_{4}({\mathbb{C}})$ by $E_{kl}$ and $\overline{E}_{kl}$ respectively. Define the following shift operators

[TABLE]

and similar operators $V_{2}$ and $V_{2}^{\prime}$ shifting $\mu_{2}$ and $\mu_{2}^{\prime}$.

Then

[TABLE]

2 Calculations

2.1. The expression for kernel.

Lemma 2.1

The kernel $K(\cdot)$ of an integral operator $T_{\mu_{1},\varepsilon_{1};\mu_{2},\varepsilon_{2}}(F)$ is given by the formula

[TABLE]

For F\in C_{0}^{\infty}\bigl{(}\mathrm{GL}_{2}({\mathbb{R}})\bigr{)} the integration is actually taken over a bounded domain.

The integral converges if $\mathop{\mathrm{Re}}\nolimits\mu_{1}>1/2$, $\mathop{\mathrm{Re}}\nolimits\mu_{2}>1/2$. For fixed $\varepsilon_{1}$, $\varepsilon_{2}$ it has a meromorphic continuation to the whole complex plane $(\mu_{1},\mu_{2})$ with poles on the hyperplanes $\mu_{1}=-1/2-k$, $\mu_{2}=-1/2-k$, where $k=0$, 1, 2, …(see, e.g., [4], §I.3).

Proof. By the definition

[TABLE]

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

[TABLE]

or

[TABLE]

The Jacobi matrix of this transformation is triangular, and the Jacobian is $|u|$. The inverse transformation is

[TABLE]

We also have $x_{11}x_{22}-x_{12}x_{21}=uw$.

After the change of variables we come to

[TABLE]

where $K(\cdot)$ is given by (2.1).

A function $F$ has a compact support in ${\mathbb{R}}^{4}\setminus\{x_{11}x_{22}-x_{12}x_{21}=0\}$. 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. $\square$

2.2. Preliminary remarks. Below $F$ denotes

[TABLE]

Also, $\partial_{11}F$, $\partial_{12}F$, etc. denote

[TABLE]

etc. Partial derivatives of $F$ are

[TABLE]

and

[TABLE]

Also, notice that

[TABLE]

2.3. A verification of the formula for $E_{14}$. It is easy to verify that the operator $e_{14}$ in C^{\infty}_{0}\bigl{(}\mathrm{GL}_{2}({\mathbb{R}})\bigr{)} is given by

[TABLE]

Therefore,

[TABLE]

(the integration is taken over ${\mathbb{R}}^{3}$ on default).

We must verify that (2.8) coincides with

[TABLE]

Below we establish two formulas

[TABLE]

[TABLE]

Considering the sum of (2.10) and (2.11) with coefficients $\frac{-1/2-\sigma+\mu_{1}}{\mu_{1}-\mu_{2}}$ and $\frac{-1/2-\sigma+\mu_{2}}{\mu_{1}-\mu_{2}}$ we get coincidence of (2.8) and (2.9); for this, we use he identities

[TABLE]

Now let us check (2.10). The following identity can be verified by a straightforward calculation (with (2.6) and (2.5)):

[TABLE]

Therefore the left-hand side of (2.10) equals to

[TABLE]

We integrate this expression by parts in the variable $w$ and come to (2.10).

To check (2.11), we verify the identity

[TABLE]

and after this integrate by parts as above.

2.4. A verification of the formula for $E_{32}$. We have

[TABLE]

Therefore,

[TABLE]

where

[TABLE]

On the other hand,

[TABLE]

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

[TABLE]

[TABLE]

Let us verify (2.14). It can be easily checked (with (2.7), (2.4), (2.5)) that

[TABLE]

We substitute this to the left-hand side of (2.14) and come to

[TABLE]

Integrating by parts, we get

[TABLE]

After a summation we come to the right-hand side of (2.14).

A proof of (2.15) is similar, we use the identity

[TABLE]

and repeat the same steps.

2.5. Table of formulas. First, we present formulas for the action of the Lie algebra $\mathfrak{g}\mathfrak{l}_{4}$ corresponding to $U_{\sigma}$, see (1.6). Denote generators of $\mathfrak{g}\mathfrak{l}_{4}$ by $e_{kl}$, where $1\leqslant k,l\leqslant 4$. Denote by $\partial_{pq}$ the partial derivatives $\frac{\partial}{\partial x_{pq}}$, 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 $\mathfrak{g}\mathfrak{l}_{2}$:

[TABLE]

b) Generators corresponding to the block $D$ also form a Lie algebra $\mathfrak{g}\mathfrak{l}_{2}$:

[TABLE]

c) Elements corresponding to the block $B$ form a 4-dimensional Abelian Lie algebra:

[TABLE]

c) Elements corresponding to the block $C$ also form a 4-dimensional Abelian Lie algebra:

[TABLE]

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,

[TABLE]

and

[TABLE]

Next,

[TABLE]

and

[TABLE]

2.6. The case of $\mathrm{GL}_{2}({\mathbb{C}})$. Notice that formulas in Subsections 1–1 for $\mathrm{SL}_{2}({\mathbb{R}})$ and in Subsections 1–1 are very similar, except the Plancherel formulas.

The analog of formula (2.1) is

[TABLE]

Its derivation is based on the same change of variables (2.2), its real Jacobian is $u\overline{u}$. A further calculation one-to-one follows the calculation for $\mathrm{GL}_{2}({\mathbb{R}})$.