Non-trivial representation of second-smallest dimension Hi,
The complex simple algebraic group $Sp_{m,\mathbb{C}}$ of $2m$-dimensional space $V$ has, for $m≥2$, an irreducible representation of dimension $m(2m−1)−1$ in a subspace of codimension $1$ of the space $\Lambda^2V$. Is it the irreducible representation of smallest dimension after $V$ itself?
Thank you.
 A: The irreducible complex representations of the  simply connected simple group $G=Sp_{r,{\mathbb C}}$ of type $C_r$,
for $r>1$, of dimension $n<{\rm dim}\ G$
are listed in the paper of Andreev, Vinberg, and Elashvili,
Table 1 (see also the Russian version). 
They are the fundamental irreducible representations $R(\pi_1)$ of dimension $2r$, $R(\pi_2)$ of dimension $2r^2-r-1$,
and, for $r=3$, $R(\pi_3)$ of dimension 14. 
For all $r\ge 2$, $r\neq 3$, we have ${\rm dim}\ R(\pi_1)=2r<2r^2-r-1={\rm dim}\ R(\pi_2)$, 
hence $R(\pi_2)$ is the nontrivial irreducible representation of second smallest dimension.
For $r=3$,  as Jim Humphreys noted, the dimensions are $6,14,14$, so ${\rm dim}\ R(\pi_2)={\rm dim}\ R(\pi_3)>{\rm dim}\ R(\pi_1)$, and 
$R(\pi_2)$ is a nontrivial irreducible representation of second smallest dimension.
A: It may be useful to expand my comments.   The question involves Lie type $C_m$ with $m \geq 2$.   Without developing Lie group or algebraic group language, it's enough to work with a simple Lie algebra over $\mathbb{C}$ of this type.    Using the standard numbering of vertices in the Dynkin diagram, let $E_i$ be the fundamental representation of highest weight $\varpi_i$ for $i= 1, \dots, m$.  Here $E_1$ is the standard module of dimension $2m$.   For the others, there are numerous classical references.   There is a thorough discussion of the construction in Bourbaki Groupes et algebres de Lie (also in English translation), Chap. VIII, $\S13$, no. 3, (IV).   In particular, the well-known dimension formula is made explicit:
$$\dim E_i = \binom{2m}{i} - \binom{2m}{i-2} \text{ for } i \geq 2$$.
Clearly $\dim E_1 > \dim E_2$.   The claim is that $\dim E_2 \geq \dim E_j$
for all $j >2$.   This should require an elementary combinatorial comparison, not involving any Lie theory, though it would be interesting to see a conceptual argument.  
Granted this inequality, Weyl's dimension formula (as already noted) will complete the desired argument for $ E_2$ being the second smallest nontrivial irreducible representation.    The formula involves a fraction, whose denominator can be ignored.  The numerator is an integral polynomial in the highest weight, which obviously grows larger as the coordinates of that weight increase relative to the $\varpi_i$.
P.S. I don't want to leave the impression that I've written down a formal proof.   It's only a proof-scheme, but should be fairly easy to complete using straightforward methods.   For the comparison between fundamental and non-fundamental weights, you'd need to look at the root system $C_m$ (say at the end of Chapters IV-VI of Bourbaki): a rough comparison of how often $\alpha_1, \alpha_2$ occur in each positive root shows for instance how the Weyl dimension for $2\varpi_1$ exceeds the dimension for $\varpi_2$, etc.    I don't recall seeing all of this written down anywhere, but if there is motivation to do so it should be elementary to complete. 
