I believe that you mean for your basis $b_\lambda$ to correspond to the basis of Schur functions in the ring of symmetric functions, in which case your $e_i$ operator corresponds to multiplication with respect to the complete symmetric function $h_i = b_{(i)} = b_i$ and your $e_i^T$ corresponds to the adjoint of multiplication by $h_i$ with respect to the Hall inner product.

Supposing this is the case, we can view the ring of symmetric functions as an algebra of operators acting on itself where we view elements in the ring of symmetric functions, $\Lambda = \mathbb{Z}[p_1, p_2, \dots]$, as polynomials in the power sums, $p_i$, and the action of $p_i$ on an element in $\Lambda$ is by multiplication.

In this case the adjoint of multiplication by $p_i$ with respect to the Hall inner product is $p_i^\perp = i \frac{d}{d p_i}$. When viewed in this way, the algebra generated by the $p_i$ and $p_i^\perp$ is the Heisenberg or oscillator algebra.

As mentioned above in S. Carnahan's answer, this can be found in Kac's *Infinite Dimensional Lie Algebras* and Kac and Raina's *Bombay Lectures*. This can also be found, from a symmetric function perspective, in Macdonald's book *Symmetric Functions and Hall Polynomials* Chapter I, Section 5, Exercise 3 which begins on page 75 (Here I am referring to the second edition).

I'm not sure what the commutation relations are between $h_i$ and $h_i^\perp$ (I am abusing notation here and using $h_i$ to mean multiplication by $h_i$), however, I can verify the suspicion stated in your response to Ben Webster's answer.

It is shown in Macdonald that for any function $f \in \Lambda$, $f(p_1, p_2, \dots)^\perp = f(p_1^\perp, p_2^\perp, \dots)$ thus $h_1^\perp = p_1^\perp = \frac{d}{dp_1}$ and so is a first order linear differential operator. In particular, chain rule works nicely and we see that $$ h_1^\perp \circ h_i = h_i \circ h_1^\perp + h_1^\perp(h_i) = h_i \circ h_1^\perp + h_{i-1} $$ where I have used $\circ$ to denote composition. Also note that $h_1^\perp(h_i) = h_{i-1}$ follows from the description of $h_1^\perp = e_1^T$ and the fact that $h_i = b_i = b_{(i)}$ in the notation of the OP.

I'm also not sure about how to express $\frac{d}{dh_i}$ in terms of $h_i$ and $h_i^\perp$, however, there may be some threads in Macdonald's book.

As for finding a nice reference for all of this, I haven't been able to find one. What I was able to piece together from the references given earlier essentially became my Master's Thesis although the purpose was somewhat different. In particular we were looking at the Bernstein operator which is related to all of this in that it arises via the Boson-Fermion Correspondence and is related to Integrable Hierarchies. It also arises in Macdonald's book I 5 Ex 29 on page 95. We took a very combinatorial approach, which can be found here. A more algebraic approach was taken by Hird, Jing and Stitzinger (here) and in particular, their approach extends to the B-type case (Schur Q Functions).