Deep learning-based synthetic aperture radar (SAR) target recognition often suffers from sparsely distributed training samples and rapid angular variations due to scattering scintillation. Thus, data-driven SAR target recognition is considered a typical few-shot learning (FSL) task. This article first reviews the key issues of FSL and provides a definition of the FSL task. A novel adversarial autoencoder (AAE) is then proposed as an SAR representation and generation network. It consists of a generator network that decodes target knowledge to SAR images and an adversarial discriminator network that not only learns to discriminate ``fake'' generated images from real ones but also encodes the input SAR image back to target knowledge. The discriminator employs progressively expanding convolution layers and a corresponding layer-by-layer training strategy. It uses two cyclic loss functions to enforce consistency between the inputs and outputs. Moreover, rotated cropping is introduced as a mechanism to address the challenge of representing the target orientation. The moving and stationary Target recognition (MSTAR) 7-target dataset is used to evaluate the AAE's performance, and the results demonstrate its ability to generate SAR images with aspect angular diversity. Using only 90 training samples with at least 25° of orientation interval, the trained AAE is able to generate the remaining 1748 samples of other orientation angles with an unprecedented level of fidelity. Thus, it can be used for data augmentation in SAR target recognition FSL tasks. Our experimental results show that the AAE could boost the test accuracy by 5.77%.