In his paper "Hecke Algebras of type $A\_n$ (Inv. Math. 1988, [EUDML link](https://eudml.org/doc/143571)) and subfactors", in section 2 "Orthogonal representations...", Wenzl takes the usual third relation of a standard generator of Hecke algebras, namely $g\_i^2=(q-1)g\_i+q$, where $q$ is a complex number that is neither $0$ nor $-1$, and says that $g\_i$ can only have at most two eigenvalues. Understandable given that the minimal polynomial of $g\_i$ must divide $x^2-(q-1)x-q$. So the possible eigenvalues must be $q,-1$. Then proceeds to say that the spectral projection belonging to $-1$ is $e\_i= \frac{q-g\_i}{q+1}$. Can someone explain to me what he means by spectral projection and how he comes up with the formula for $e\_i$?



Spectral projection of an eigenvalue associated to a generator of Hecke algebras
Since $e\_i$ kills any eigenvectors with eigenvalue $q$ (clear) and squares to itself (due to the quadratic relation for $g\_i$ and normalisation), it is a projector onto the $-1$-eigenspace of $g\_i$. For the usual (reducible) representation of $H\_n(q)$ on $V^{\otimes n}$, where $g\_i$ becomes the simple transposition $s\_i$ as $q\to 1$, the element $e\_i$ acts as the antisymmetriser in the $i$ and $i+1$st factors of $V^{\otimes n}$. So one may think of $e\_i$ as a $q$-antisymmetriser.


For comparison, the spectral projector onto the $q$-eigenspace of $g\_i$ would be $-(1+g\_i)/(q+1)$. Note that these two projectors orthogonal (again due to the quadratic relation) and add to the identity. One can think of this as the corresponding $q$-symmetriser.


(If $g\_i$ acts on $V^{\otimes n}$ and $V \cong \mathbb{C}^2$, then the left- and right-hand sides of Wenzl's relation (H1) both vanish, and the $H\_n(q)$-representation on $V^{\otimes n}$ factors through a representation of the Temperley--Lieb algebra, with generators $e\_i$ up to normalisation depending on conventions.)


I hope this helps.


PS. Nowadays the Hecke generators are usually denoted by $T\_i$ rather than $g\_i$. The notation $e\_i$ for (something proportional to) $q-T\_i$ is standard.


