To any complex simple Lie algebra $\mathfrak g$ of rank $n$ there corresponds a set of $n$ natural numbers $\{e_1,,e_2,,\dots,,e_n\}$ called “exponents”. They have many remarkable properties. E.g., the degree of $\mathfrak g$-invariant polynomials are $m_i=e_i+1$, and degree of $\mathfrak g$-invariant elements in $\wedge(\mathfrak g$ are $M_i= 2e_i+1$. About 50 years ago B.Kostant introduced generalized exponents for any finite-dimensional irreducible representation $\pi_\la$ of $\g$. The number of these exponents is equal to $m_\la(0)$, the multiplicity of the zero weight in $\pi_\la$. But until now there is no effective way to compute these numbers. Ten years ago I introduced a new sort of associative algebras, related to a pair $(\mathfrak g,,\pi_\la)$ in a hope that they will be useful in study of generalized exponents. In the talk I explain the last results in this direction and give a simple description of some classical family algebras.