On O m ×G L n Highest Weight Vectors

Let ℂm,n be the vector space of m×n complex matrices and P(ℂm,n) be the algebra of complex-valued polynomials on ℂm,n. Let GL _m ×GL_n act on P(ℂm,n) by pre-and post-multiplication as follows: $$\left( {{g_1},{g_2}} \right)f\left( x \right) = f\left( {g_1^{ - 1}x{g_2}} \right)$$ where x ∈ ℂm,n, (g₁,g₂) ∈ GL _m ×GL_n,f ∈ P(ℂm,n). We choose a system of coordinates on ℂm,n as follows:$$\left[ {\begin{array}{*{20}{c}}{{x_{11}}}{{x_{12}}} \ldots {{x_{1n}}} \\{{x_{21}}}{{x_{22}}} \ldots {{x_{2n}}} \\\vdots \vdots \cdots \vdots \\{{x_{m1}}}{{x_{m2}}} \cdots {{x_{mn}}}\end{array}} \right]$$