Kirillov Polynomials for the Exceptional Lie Algebra \(\mathfrak g_{2}\)

As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra \(\mathfrak g_2\) ${\mathfrak{g}}_2$ . In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.