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 ${\mathfrak{g}}_2$. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.