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UDC 519.46
MATHEMATICS
I. A. ZHIGULIN
ON SOME INVARIANTS OF THE GROUP OF UNITRIANGULAR MATRICES IN THE SPACE DUAL TO THE SPACE OF ITS LIE ALGEBRA
(Presented by Academician P. S. Novikov, 21 V 1965)
A. A. Kirillov in the paper \((^1)\) gave a method for describing irreducible unitary representations of simply connected nilpotent Lie groups, which consists in the following. Let \(\mathfrak G\) be a simply connected nilpotent Lie group, \(G\) its Lie algebra, \(\rho\) the adjoint representation of \(\mathfrak G\) in \(G\). Denote by \(G'\) the space dual to \(G\), and by \(\rho'\) the representation of \(\mathfrak G\) in \(G'\) dual to \(\rho\). Then the irreducible unitary representations of \(\mathfrak G\) correspond one-to-one to the orbits in \(G'\) with respect to \(\rho'(\mathfrak G)\).
In the present note the problem of classifying the orbits for the group of unitriangular matrices of order \(n\) is considered.
- Let \(\mathfrak G_n\) be the group of upper triangular matrices of order \(n\) with ones on the main diagonal, and \(G_n\) the Lie algebra of this group. We identify the space \(G_n'\), dual to \(G_n\), with the space of lower triangular matrices. An element \(x^{ij}\) \((i>j)\) of a matrix \(x \in G_n'\) will be called an element belonging to the \(k\)-th oblique row if \(i-j=k\). In what follows we shall denote the elements of the matrix \(x\) by \(x^{i+k,i}\). The basis \(\mathcal B_0\) of the space \(G_n'\) consists of the coordinate dyads \(e_{i+k,i}\) \((i,\ k=1,2,\ldots,n-1;\ i+k\le n)\), i.e. of matrices with ones at the intersection of the \(i\)-th column with the \(k\)-th oblique row and zeros in the remaining places. In the space \(G_n'\) there is defined a representation \(\rho'(\mathfrak G_n)\) of the group \(\mathfrak G_n\) by the formula
\[ \rho'(g)x = [gxg^{-1}]_{\mathrm n}, \tag{1} \]
where \(g \in \mathfrak G_n\). In (1) the symbol \([gxg^{-1}]_{\mathrm n}\) denotes the lower triangular matrix in which the elements lying below the main diagonal coincide with the corresponding elements of the matrix \(gxg^{-1}\).
Denote \(g=\|g_{\alpha\beta}\|\) and \(g^{-1}=\|h_{\alpha\beta}\|\). The elements of the basis \(\mathcal B_0\) are transformed according to the following law:
\[ \widetilde e_{i+k,i} = \rho'(g)e_{i+k,i} = \sum_{\alpha=i+1}^{i+k} \sum_{\beta=i}^{i+k-1} g_{\alpha,i+k}\,h_{i\beta}\,e_{\alpha\beta}. \tag{2} \]
Let \({}'G_k^i\) be the subspace in \(G_n'\) consisting of matrices \(x=\|x^{\alpha\beta}\|\) for which \(x^{\alpha\beta}=0\) for \(\alpha>i+k\) and \(\beta<i\). It follows from (2) that \({}'G_k^i\) is invariant with respect to \(\rho'(\mathfrak G_n)\) and contains an orbit \(\Omega_k^i\) which is a transitive submanifold in it. In what follows we shall call the subspace \({}'G_k^i\) the \(\binom{i}{k}\)-th block. The coordinates \(\widetilde x^{\alpha\beta}\) of the image \(\widetilde x\) of an element \(x \in {}'G_k^i\) under transformations from \(\rho'(\mathfrak G_n)\) are found by the formula
\[ \widetilde x^{\alpha\beta} = \sum_{\gamma=\alpha}^{i+k} \sum_{\delta=i}^{\beta} g_{\alpha\gamma}x^{\gamma\delta}h_{\delta\beta}. \tag{2a} \]
It is not hard to show that the orbit \(\Omega_k^i\), contained in \({}'G_k^i\), is for it an orbit of general position (see (1), p. 106). We arrive at the following result:
a) By specifying the element \(e_{i+k,i}\) of the basis \(\mathcal B_0\) of the space \(G_n'\), one singles out in the latter the subspace \({}'G_k^i\), invariant with respect to \(\rho'(\mathfrak G_n)\) and containing the \(\binom{i}{k}\)-th class of orbits. The orbit \(\Omega_k^i\) from the \(\binom{i}{k}\)-th class is an algebraic surface defined by the equations:
\[ {}_\nu\Delta_k^i=\mathrm{const}\ne 0 \quad (\nu=1,2,\ldots,[(k+1)/2]), \tag{3} \]
where \({}_\nu\Delta_k^i\) is the determinant of the minor of order \(\nu\) of the left lower corner of the \(\binom{i}{k}\)-th block.
\[ \dim \Omega_k^i=k(k+1)/2-[(k+1)/2]. \]
In what follows, the element \(e_{i+k,i}\) of the basis \(\mathcal B_0\), defining the \(\binom{i}{k}\)-th class of orbits, will be called the leading vector of the class of orbits.
- Let \(m\) elements of the basis \(\mathcal B_0\) be taken,
\[ e_{i_1+k_1,i_1},\quad e_{i_2+k_2,i_2},\ldots;\quad e_{i_m+k_m,i_m}, \tag{4} \]
where the numbers \(i_\varepsilon\) and \(k_\varepsilon\) satisfy the conditions:
\[ 1\le i_1<i_2<\cdots<i_m\le n-1, \]
\[ i_\varepsilon+k_\varepsilon<i_{\varepsilon+1}+k_{\varepsilon+1} \quad(\varepsilon=1,2,\ldots,m-1),\quad i_m+k_m\le n. \tag{5} \]
From the transformation law for coordinate vectors it follows that \(m\) leading vectors (4) single out in \(G_n'\) a subspace
\[ {}'G_{k_1 k_2\ldots k_m}^{i_1 i_2\ldots i_m} = \bigcap_{\alpha=1}^{m}{}'G_{k_\alpha}^{i_\alpha}, \]
invariant with respect to \(\rho'(\mathfrak G_n)\) and containing the
\[ \binom{i_1\, i_2\, \ldots\, i_m}{k_1\, k_2\, \ldots\, k_m} \]
-th class of orbits. The orbit
\[ \Omega_{k_1 k_2\ldots k_m}^{i_1 i_2\ldots i_m}, \]
as a transitive manifold in
\[ {}'G_{k_1 k_2\ldots k_m}^{i_1 i_2\ldots i_m}, \]
is singled out in it by a system of functions of the coordinates \(x^{\alpha\beta}\) of an element
\[ x\in {}'G_{k_1 k_2\ldots k_m}^{i_1 i_2\ldots i_m}, \]
invariant with respect to \(\rho'(\mathfrak G_n)\). Here two systems of such invariant functions are indicated for \(m=2\).
We first indicate two special cases, interesting in that these are the only possibilities when the orbits are linear subspaces in \(G_n'\). For \(k_1=k_2=\cdots=k_m=1\) we obtain the class of zero-dimensional orbits (the orbit is a point in \(G_n'\)).
For the case when the leading vectors (4) satisfy the conditions
\[ k_1=k_2=\cdots=k_m=2,\quad i_{\varepsilon+1}=i_\varepsilon+1 \quad(\varepsilon=1,2,\ldots,m-1), \tag{6} \]
the following assertion is true:
\(\beta)\) \(m\) leading vectors (4) under conditions (6) determine the
\[ \binom{i_1\, i_2\, \ldots\, i_m}{k_1\, k_2\, \ldots\, k_m} \]
-th class of orbits. The orbit
\[ \Omega_{k_1 k_2\ldots k_m}^{i_1 i_2\ldots i_m} \]
is:
a) for even \(m\), an \(m\)-dimensional plane defined by the equations
\[ x^{\alpha+2,\alpha}=\mathrm{const}\ne 0 \quad(\alpha=1,2,\ldots,m), \]
\[ A_1x^{i_1+1,i_1}+A_2x^{i_3+1,i_3}+\cdots+ A_{m/2+1}x^{m+2,i_m+1}=B, \tag{7} \]
where \(A_1,A_2,\ldots,A_{m/2+1},B\) are constants determined by the choice of the vector
\[ x\in{}'G_{k_1 k_2\ldots k_m}^{i_1 i_2\ldots i_m}. \]
b) for odd \(m\), an \((m+1)\)-dimensional plane defined by the equations
\[ x^{\alpha+2,\alpha}=\operatorname{const}\ne 0 \qquad (\alpha=1,2,\ldots,m). \]
- Let us return again to the case \(m=1\). From the elements of the matrix \(x\in {}'G_k^i\) we find two series of determinants. To this end, in the matrix \(x\in{}'G_k^i\) we single out the \((i+p)\)-th row and, from the elements that are not identically zero, form determinants \({}^{\alpha}_{p}D_k^i\) of order \(p\) \((p=2,3,\ldots,[(k+1)/2])\) \((\alpha=i+p,i+p+1,\ldots,i+k-p+1)\) in the following way: the first row of the determinant consists of the first \(p\) elements of the \(\alpha\)-th row of the matrix \(x\), while the remaining \(p-1\) rows are the last \(p-1\) rows of the matrix \(x\), taken in the same order as in the matrix \(x\), and containing the first \(p\) elements. Thus the determinants \({}^{\alpha}_{p}D_k^i\) have the form
\[ {}^{\alpha}_{p}D_k^i= \left| \begin{array}{cccc} x^{\alpha,i} & x^{\alpha,i+1} & \ldots & x^{\alpha,i+p}\\ x^{i+k-(p-1),i} & x^{i+k-(p-1),i+1} & \ldots & x^{i+k-(p-1),i+p}\\ x^{i+k-(p-2),i} & x^{i+k-(p-2),i+1} & \ldots & x^{i+k-(p-2),i+p}\\ \ldots & \ldots & \ldots & \ldots\\ x^{i+k,i} & x^{i+k,i+1} & \ldots & x^{i+k,i+p} \end{array} \right| \tag{8} \]
Similarly, we define the second series of determinants of order \(p\), \({}^{\alpha}_{p}Q_k^i\) \((\alpha=i+p-1,i+p,\ldots,i+k-p)\). To do this, in the matrix \(x\in{}'G_k^i\) we single out the last \(p\) rows. Then, in the determinant \({}^{\alpha}_{p}Q_k^i\), the \(p-1\) columns are the first \(p-1\) columns of the selected \(p\) rows, and the \(p\)-th column is the \(\alpha\)-th column in the selected \(p\) rows. Thus
\[ {}^{\alpha}_{p}Q_k^i= \left| \begin{array}{ccccc} x^{i+k-p,i} & x^{i+k-p,i+1} & \ldots & x^{i+k-p,i+p-1} & x^{i+k-p,\alpha}\\ x^{i+k-p+1,i} & x^{i+k-p+1,i+1} & \ldots & x^{i+k-p+1,i+p-1} & x^{i+k-p+1,\alpha}\\ \ldots & \ldots & \ldots & \ldots & \ldots\\ x^{i+k,i} & x^{i+k,i+1} & \ldots & x^{i+k,i+p-1} & x^{i+k,\alpha} \end{array} \right|. \tag{9} \]
It is easy to see that for \(p=[(k+1)/2]\)
\[ {}^{\alpha}_{p}D_k^i={}^\alpha_p Q_k^i={}_{[(k+1)/2]}\Delta_k^i. \]
Moreover, by definition, we put
\[ {}^{\alpha}_{1}D_k^i=x^{\alpha i},\qquad {}^{\alpha}_{1}Q_k^i=x^{i+k,\alpha}. \]
For the determinants \({}^{\alpha}_{p}D_k^i\) and \({}^{\alpha}_{p}Q_k^i\), denote by \({}^{\alpha}_{p}\widetilde D_k^i\) and \({}^{\alpha}_{p}\widetilde Q_k^i\) their images under the transformations from \(\rho'(\mathfrak G_n)\). Then
\[ {}^{\alpha}_{p}\widetilde D_k^i = \sum_{\beta=\alpha}^{i+k-p-1} g_{\alpha\beta}\,{}^{\beta}_{p}D_k^i, \tag{10} \]
\[ {}^{\alpha}_{p}\widetilde Q_k^i = \sum_{\beta=i+p-1}^{\alpha} {}^{\beta}_{p}Q_k^i\,h_{\beta\alpha}. \tag{11} \]
Now one can describe the first type of functions of the coordinates \(x^{\alpha\beta}\) of an element
\(x\in{}'G_{k_1k_2}^{\,i_1i_2}\), invariant with respect to \(\rho'(\mathfrak G_n)\). The subspace
\({}'G_{k_1k_2}^{\,i_1i_2}\) consists of matrices \(x=\|x^{\alpha\beta}\|\) in which \(x^{\alpha\beta}=0\) for all \(\alpha\) when \(i_1>\beta\), for \(i_2+k_2<\alpha<i_1+k_1\) when \(i_1\le \beta<i_2\), and for all \(\beta\) when \(\alpha>i_2+k_2\), while for the remaining values of \(\alpha\) and \(\beta\) the \(x^{\alpha\beta}\) are arbitrary.
The subspace \({}'G_{k_1k_2}^{\,i_1i_2}\) can be represented as the sum of two subspaces
\[ {}'G_{k_1k_2}^{\,i_1i_2} = {}'G_{k_1}^{\,i_1}+{}'G_{k_2}^{\,i_2}. \tag{12} \]
If \({}'G_{k_1}^{\,i_1}\cap{}'G_{k_2}^{\,i_2}=0\), then the subspace \({}'G_{k_1k_2}^{\,i_1i_2}\) decomposes into the direct sum of the subspaces \({}'G_{k_1}^{\,i_1}\) and \({}'G_{k_2}^{\,i_2}\), which leads to the decomposition of the orbit \(\Omega_{k_1k_2}^{\,i_1i_2}\) into the orbits \(\Omega_{k_1}^{\,i_1}\) and \(\Omega_{k_2}^{\,i_2}\).
Let \({}'G_{k_1}^{\,i_1}\cap {}'G_{k_2}^{\,i_2}\ne 0\). In the matrix \(x\in {}'G_{k_1k_2}^{\,i_1i_2}\) we single out \(i_2-i_1\) first columns, beginning with the \(i_1\)-st, and \(i_2+k_2-i_1-k_1\) last rows, beginning with the \((i_1+k_1+1)\)-st, and define two systems of determinants of orders \(p\) and \(r\), respectively:
\[ {}_{p}^{\alpha}D_{k_1}^{\,i_1}\quad (\alpha=i_1+p,\ i_1+p+1,\ldots,\ i_1+k_1-p+1), \qquad {}_{r}^{\alpha}Q_{k_2}^{\,i_2}\quad (\alpha=i_2+r-1,\ i_2+r,\ldots,\ i_2+k_2-r), \tag{13} \]
where \(p=1,2,\ldots,i_2-i_1\) if \(i_2-i_1<[(k_1+1)/2]\), and \(p=1,2,\ldots,[(k_1+1)/2]\) if \(i_2-i_1\ge[(k+1)/2]\); \(r=1,2,\ldots,i_2+k_2-i_1-k_1\) if \(i_2+k_2-i_1-k_1<[(k_2+1)/2]\), and \(r=1,2,\ldots,[(k_2+1)/2]\) if \(i_2+k_2-i_1-k_1\ge[(k_2+1)/2]\).
The following assertion holds:
\(\gamma)\) The expression
\[ \sum_{\alpha=i_2+r-1}^{\,i_1+k_1-p+1} {}_{r}^{\alpha}Q_{k_2}^{\,i_2}\cdot {}_{p}^{\alpha}D_{k_1}^{\,i_1}, \tag{14} \]
formed from the determinants \({}_{r}^{\alpha}D_{k_1}^{\,i_1}\) and those determinants \({}_{r}^{\alpha}Q_{k_2}^{\,i_2}\) whose order \(r\), for the given \(p\), satisfies the condition
\(i_2+r-1\le i_1+k_1-p+1\), remains invariant under all transformations from \(\rho'(\mathfrak{G}_n)\).
- For the same case \(m=2\) we carry out the following constructions: we complete the subspace \({}'G_{k_1k_2}^{\,i_1i_2}\) to \({}'G_{i_2+k_2-i_1}^{\,i_1}\), replacing the elements \(x^{\alpha\beta}\)
\((i_1+k_1<\alpha\le i_2+k_2;\ i_1\le \beta<i_2)\), identically equal to zero, by elements \(\bar{x}^{\alpha\beta}\) with the same values \(\alpha\) and \(\beta\) from the square of the matrix \(x\), and leaving the remaining ones unchanged. In the subspace \({}'G_{i_2+k_2-i_1}^{\,i_1}\) the same subgroup \({}'\rho_{k_1k_2}^{\,i_1i_2}(\mathfrak{G}_n)\) of the group \(\rho'(\mathfrak{G}_n)\) acts as in the subspace \({}'G_{k_1k_2}^{\,i_1i_2}\). The elements \(\bar{x}^{\alpha\beta}\)
\((i_1+k_1<\alpha\le i_2+k_2;\ i_1\le \beta<i_2)\) under the action of transformations from \({}'\rho_{k_1k_2}^{\,i_1i_2}(\mathfrak{G}_n)\) transform according to the formula
\[ \bar{x}^{\alpha\beta} = \sum_{\gamma=\alpha}^{\,i_2+k_2} \sum_{\delta=i_1}^{\,\beta} g_{\alpha\gamma}\bar{x}^{\gamma\delta}h_{\delta\beta}. \tag{15} \]
From formula (15) it follows directly:
\(\delta)\) In the subspace \({}'G_{k_1k_2}^{\,i_1i_2}\) the functions
\[ {}_{\nu}\Delta_{i_2+k_2-i_1}^{\,i_1}, \tag{16} \]
where \({}_{\nu}\Delta_{i_2+k_2-i_1}^{\,i_1}\) is the determinant of the minor of order \(\nu\) of the lower left corner of the block
\(\left(\begin{array}{c}
i_1\\
i_2+k_2-i_1
\end{array}\right)\), are invariant with respect to \(\rho'(\mathfrak{G}_n)\), and
\[ \nu= \begin{cases} 1,2,\ldots,i_2+k_2-i_1-k_1, & \text{if } k_1<k_2,\\ 1,2,\ldots,i_2-i_1, & \text{if } k_1\ge k_2, \end{cases} \]
and always \(\nu\le[(i_2+k_2-i_1+1)/2]\).
It is easy to see that in \({}'G_{k_1k_2}^{\,i_1i_2}\), besides invariants of types (14) and (16), there are invariants of type (3), defined for
\({}'G_{k_1}^{\,i_1}\subset{}'G_{k_1k_2}^{\,i_1i_2}\) and
\(G_{k_2}^{\,i_2}\subset{}'G_{k_1k_2}^{\,i_1i_2}\), with the known restrictions on \(\nu\).
In conclusion, the author expresses sincere gratitude to G. B. Gurevich for his constant attention and valuable advice.
Moscow State
Pedagogical Institute
named after V. I. Lenin
Received
14 V 1965
REFERENCES
- A. A. Kirillov, UMN, 17, no. 4 (1962).